Subcomplete Forcing and ℒ-Forcing

July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 1 Subcomplete Forcing and L-Forcing Ronald Jensen ABSRACT In his book Proper Forcing (1982) Shelah introduced three classes of forcings (complete, proper, and semi-proper) and proved a strong iteration theorem for each of them: The first two are closed under countable support iterations. The latter is closed under revised countable support iterations subject to certain standard restraints. These theorems have been heavily used in modern set theory. For instance using them, one can formulate “forcing axioms” and prove them consistent relative to a supercompact cardinal. Examples are PFA, which says that Martin’s axiom holds for proper forcings, and MM, which says the same for semiproper forcings. Both these axioms imply the negation of CH. This is due to the fact that some proper forcings add new reals. Complete forcings, on the other hand, not only add no reals, but also no countable sets of ordinals. Hence they cannot change a cofinality to ω. Thus none of these theories enable us e.g. to show, assuming CH, that Namba forcing can be iterated without adding new reals. More recently we discovered that the three forcing classes mentioned above have natural generalizations which we call “subcomplete”, “subproper” and “semisubproper”. It turns out that each of these is closed under Revised Countable Support (RCS) iterations subject to the usual restraints. The first part of our lecture deals with subcomplete forcings. These forcings do not add reals. Included among them, however, are Namba forcing, Prikry forcing, and many other forcings which change cofinalities. This gives a positive solution to the above mentioned iteration problem for Namba forcing. Using the iteration theorem one can also show that the Subcomplete Forcing Axiom (SCFA) is consistent relative to a supercompact cardinal. It has some of the more striking consequences of MM but is compatible with CH (and in fact with ♦). (Note: Shelah was able to solve the above mentioned iteration problem for Namba forcing by using his ingenious and complex theory of “I-condition forcing”. The relationship of I-condition forcing to subcomplete forcing remains a mystery. There are, however, many applications of subcomplete forcing which have not been replicated by I-condition forcing.) In the second part of the lecture, we give an introduction to the theory of “LForcings”. We initially developed this theory more than twenty years ago in order to force the existence of new reals. More recently, we discovered that there is an interesting theory of L-Forcings which do not add reals. (In fact, if we assume CH +2ω1 = ω2 , then Namba forcing is among them.) Increasingly we came to feel that there should be a “natural” iteration theorem which would apply to a large class of these forcings. This led to the iteration theorem for subcomplete forcing. July 21, 2012 15:2 2 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing Combining all our methods, we were then able to prove: (1) Let κ be a strongly inaccessible cardinal. Assume CH. There is a subcomplete forcing extension in which κ becomes ω2 and every regular cardinal τ ∈ (ω1 , κ) acquires cofinality ω. (2) Let κ be as above, where GCH holds below κ. Let A ⊂ κ. There is a subcomplete forcing extension in which: – κ becomes ω2 ; – If τ ∈ (ω1 , κ) ∩ A is regular, then it acquires cofinality ω; – If τ ∈ (ω1 , κ)\A is regular, then it acquires cofinality ω1 . We will not be able to fully prove these theorems in our lectures, but we hope to develop some of the basic methods involved. jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen Contents 0. Preliminaries 1 1. Admissible sets 1.1 1.2 1.3 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ill founded ZF− models . . . . . . . . . . . . . . . . . . . . . . . . Primitive Recursive Set Functions . . . . . . . . . . . . . . . . . . 11 16 17 2. Barwise Theory 19 3. Subcomplete Forcing 29 3.1 3.2 3.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liftups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 33 42 4. Iterating subcomplete forcing 51 5. L-Forcing 69 6. Examples 81 6.1 6.2 6.3 6.4 Example 1 . . . . . . . . . . . . Example 2 . . . . . . . . . . . . Example 3 . . . . . . . . . . . . The extended Namba problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 84 87 88 95 Bibliography 3 July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Chapter 0 Preliminaries ZF− (“ZF without power set”) consists of the axioms of extensionality and foundation together with: S (1) ∅, {x, y}, x are sets. (2) (Axiom of Subsets or “Aussonderungsaxiom”) x ∩ {z | ϕ(z)} is a set. (3) (Axiom of Collection) V W V W V W x y ϕ(x, y) → u v x ∈ u y ∈ v ϕ(x, y) (4) (Axiom of Infinity) ω is a set. Note (3) implies the usual replacement axiom, but cannot be derived from it without the power set axiom. ZFC− is ZF− together with the strong form of the axiom of choice: (5) Every set is enumerable by an ordinal. Note The power set axiom is required to derive (5) from the weaker forms of choice. The Levy hierarchy of formulae is defined in the usual way: Σ0 formulae are the formulae containing only bounded quantification – i.e. Σ0 = the smallest set of formulae containing the primitive formulae and closed under sentential operations and bounded quantification: V W x ∈ y ϕ, x∈yϕ V V W W (where x ∈ y ϕ = x(x ∈ y → ϕ) and x ∈ y ϕ = x(x ∈ y ∧ ϕ)). (In some contexts it is useful to introduce bounded quantifiers as primitive signs rather than defined operations.) W We set: Π0 = Σ0 . Σn+1 formulae are then the formulae of the form x ϕ, V where ϕ is Πn . Similarly Πn+1 formulae have the form x ϕ, where ϕ is Σn . A relation R on the model A is called Σn (A) (Πn (A)) iff it is definable over A by a Σn (Πn ) formula. 1 jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 2 Subcomplete Forcing and L-Forcing R is Σn (A) (Πn (A)) in the parameters p1 , . . . , pm iff it is Σn (Πn ) definable in the parameters p1 , . . . , pn ∈ A. It is Σn (A) (Πn (A)) iff it is Σn (Πn ) definable in some parameters. It is ∆n (A) iff it is Σn (A) and Πn (A). ⋆ ⋆ ⋆ ⋆ ⋆ x or card(x) denotes the cardinality of x. (We reserve the notation |x| for other uses.) If r is a well ordering or a set of ordinals, then otp(r) denotes its order type. crit(f ) is the critical point of the function f (i.e. α = crit(f ) ↔ (f ↾ α = id ∧ f (α) > α). F ′′ A is the image of A under the function (or relation) F . rng(R) is the range of the relation R. dom(R) is the domain of the relation R. TC(x) is the transitive closure of x, Hα = {x | TC(x) < α}. Boolean Algebras and Forcing The theory of forcing can be developed using ”sets of conditions“ or complete Boolean algebras. The former is most useful when we attempt to devise a forcing for a specific end. The latter is more useful when we deal with the general theory of forcing, as in the theory of iterated forcing. We adopt here an integrated approach which begins with Boolean algebras. By a Boolean algebra we mean a partial ordering B = h|B|, cB i with maximal and minimal elements 0, 1, lattice operations ∩, ∪ defined by: a ⊂ (b ∩ c) ←→ (a ⊂ b ∧ a ⊂ c) (b ∪ c) ⊂ a ←→ (b ⊂ a ∧ c ⊂ a) and a complement operation ¬ defined by: a ⊂ ¬b ←→ a ∩ b = 0, satisfying the usual Boolean equalities. We call B a complete Boolean algebra if, in T S addition, for each X ⊂ B there are operations B X, B X defined by: TB V a⊂ X ←→ b ∈ X a ⊂ b, SB V X ⊂ a ←→ b ∈ X b ⊂ a, s.t. a∩ [ b∈I b= [ b∈I (a ∩ b), a∪ \ b∈I = \ (a ∩ b). b∈I We shall generally write ’BA’ for ’Boolean algebra’. We write A ⊆ B to mean that A, B are BA’s, A is complete, and A is completely contained in B – i.e. TB TA SB SA X= X, X= X for X ⊂ A. T If A ⊆ B and b ∈ B, we define h(b) = hA,B (b) by: h(b) = {a ∈ A | b ⊂ a}. Thus: jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 3 Preliminaries S S • h( bi ) = h(bi ) if bi ∈ B for i ∈ I. i i • h(a ∩ b) = a ∩ h(b) if a ∈ A. • b = 0 ↔ h(b) = 0 for b ∈ B. ⋆ ⋆ ⋆ ⋆ ⋆ If B is a complete BA, we can form the canonical maximal B-valued model VB . The elements of VB are called names and there is a valuation function ϕ → [[ϕ]]B attaching to each statement ϕ = ϕ′ (t1 , . . . , tn ) a truth value in B. (Here ϕ is a ZFC formula and t1 , . . . , tn are names.) All axioms of ZFC have truth value 1 (assuming ZFC). The sentential connectives are interpreted by: [[ϕ ∧ ψ]] = [[ϕ]] ∩ [[ψ]]; [[ϕ ∨ ψ]] = [[ϕ]] ∪ [[ψ]]; [[ϕ → ψ]] = [[ϕ]] ⇒ [[ψ]], where (a ⇒ b) =Df ¬a ∪ b; [[¬ϕ]] = ¬[[ϕ]]. The quantifiers are interpreted by: \ V [[ϕ(x)]], [[ v ϕ(v)]] = [ W [[ϕ(x)]]. [[ v ϕ(v)]] = x∈VB x∈VB If u ⊂ VB is a set and f : u → B, then there is a name x ∈ VB s.t. [ [[y ∈ x]] = [[y = z]] ∩ f ( ) z∈u B for all y ∈ V . Conversely, for each x ∈ VB there is a set ux ⊂ VB s.t. [ [[y = z]] ∩ [[z ∈ x]]. [[y ∈ x]] = z∈ux We can, in fact, arrange things s.t. {hz, xi | z ∈ ux } is a well founded relation. If ◦ U ⊂ VB is a class j and A : U → B, we may add to the language a predicate A ◦ S [[x = z]] ∩ A(z). We inductively define for each x ∈ V interpreted by: [[Ax]] = z∈u a name x̌ by: [[y ∈ x̌]] = [ [[y = ž]], z∈x and a predicate V̌ by: [[y ∈ V̌]] = [ [[y = ž]]. τ ∈V ∼ If σ : A ↔ B is an isomorphism, then we can define an injection σ ∗ : VA → VB as follows: Let R = {hz, xi | z ∈ ux } be the above mentioned well founded relation for VA . By R-induction we define σ ∗ (x), picking σ ∗ (x) to be a w ∈ VB s.t. [ [[y = σ ∗ (z)]]B ∩ σ([[z ∈ x]]A ). [[y ∈ w]]B = z∈ux Then: July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 4 (1) Subcomplete Forcing and L-Forcing σ([[ϕ(~x )]]A ) = [[ϕ(σ ∗ (~x ))]]B for all ZFC formulae and all x1 , . . . , xn ∈ VA . If σ : A → B is a complete embedding ∼ (i.e. σ : A ↔ A′ ⊆ B for some A′ ), then σ ∗ can be defined the same way, but (1) then holds only for Σ0 formulae. In such contexts it is often useful to take VB as a B-valued identity model, meaning that [[x = y]] = 1 −→ x = y for x, y ∈ VB . (If VB does not already have this property, we can attain it by factoring.) If ∼ σ : A ↔ B and VA , VB are identity models, then σ ∗ is bijective (and is, in fact, an isomorphism of hVA , I A , E A i onto hVB , I B , E B i, where I = (x, y) = [[x = y]], E(x, y) = [[x ∈ y]]). Another advantage of identity models in that {z | [[z ∈ x]] = 1} is then a set, rather than a proper class. There are many ways to construct a maximal B-valued model VB and we can take its elements as being anything we want. Noting that A ⊆ B means that id ↾ A is a complete embedding, it is useful, when dealing with such a pair A, B, to arrange that VA ⊂ VB and (id ↾ A)∗ = id ↾ VA . (We express this by: VA ⊆ VB .) The forcing relation B is defined by: b ϕ ←→Df (b 6= 0 ∧ b ⊂ [[ϕ]]). We also set: ϕ ↔Df [[ϕ]] = 1. Now suppose that W is an inner model of ZF and B ∈ W is complete in the sense of W . We can form W B internally in W , and it turns out that all ZF axioms are true in W B . (If W satisfies ZFC, then ZFC holds in W B .) W could also be a set rather than a class. If W is only a model of ZF− , we can still form W B , which will then model ZF− (or ZFC− if W models ZFC− ). (In this case, however, we may not be able – internally in W – to factor W B to an identity model.) We say that G ⊂ B is B-generic over W iff G is an ultrafilter on B which respects all intersections and unions of X ⊂ B s.t. X ∈ W – i.e. T V S W x ∈ G ←→ b ∈ x b ∈ G, x ∈ G ←→ b ∈ X b ∈ G. If G is generic, we can form the generic extension W [G] of W by: W [G] = {xG | x ∈ W G }, where xG = {z G | z ∈ ux ∧ [[z ∈ x]] ∈ G}. Then W ⊂ W [G], since x̌G = x (by G-induction on x ∈ W ). Then: W G W [G]  ϕ(xG b ∈ G b ϕ(x1 , . . . , xn ). 1 , . . . , xn ) ←→ If we suppose, moreover, that for every b ∈ B \ {0} there is a generic G ∋ b (e.g. if ϕ(B) ∩ W is countable), then: b G ϕ(x1 , . . . , xn ) ←→ (W [G]  ϕ(xG 1 , . . . , xn ) for all generic G ∋ b). If B is complete in V we shall often find it useful to work in a mythical universe in which: jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 5 Preliminaries (∗) V is an inner model and for every b ∈ B \ {0} there is a G ∋ b which is B-generic over V. This is harmless, since if C collapsed 2B to ω, then (∗) holds of V̌, B̌ in VC . We ◦ note that there is a G ∈ VB s.t. ◦ ◦ ◦ G ⊂ B̌ and [[b̌ ∈ G]] = b for b ∈ B. (G is in fact ◦ unique if VB is an identity model.) If then G is B-generic over V, we have GG = G. Thus ◦ ◦ G is B̌-generic over V̌. We call G the canonical B-generic name. ◦ If our language contains predicates A other than 0, ∈, we set: ◦ ◦ AG = {xG | [[x ∈ A]] ∈ G}. Since [[x ∈ V̌ ]] = S [[x = ž]], we get: z∈V V̌G = {ž G | z ∈ V} = V. Sets of Conditions By a set of conditions we mean P = h|P|, ≤P i s.t. ≤=≤p is a transitive relation on |P|. (Notationally we shall not distinguisch between P and |P|.) We say that two W conditions p, q are compatible (pkq) if r r ≤ p, q. Otherwise they are incompatible (p⊥q). For each set of conditions P there is a canonical complete BA over P (BA(P)) defined as follows: For X ⊂ P set: V ¬X = {q | p ∈ X p⊥q}. Then X ⊂ ¬¬X and ¬¬¬X = ¬X. Hence ¬¬ is a hull operator on P(P). Set |B| = {X ⊂ P | X = ¬¬X}. Then BA(P) = h|B|, ci, where c is the ordinary inclusion relation on |B|. B = BA(P) is then a complete BA with the complement operation ¬ and intersection and union operations given by: TB T SB S X = X, X = ¬¬ X. V W We say that ∆ ⊂ P is dense in P iff p ∈ P q ⊆ p, q ∈ ∆. ∆ is predense in P iff V W p ∈ P q (qkp and q ∈ ∆). (In other words, the closure of ∆ under ≤ is dense in P.) T Set: [p] = ¬¬{p} for p ∈ P (i.e. [p] = {a ∈ BA(P) | a ⊃ b}). The forcing relation for P is defined by: p ϕ ←→Df [p] ⊂ [[ϕ]]. If P ∈ W and W is a transitive model of ZF, we say that G is P-generic over W iff the following hold: • If p, q ∈ G, then pkq. • If p ∈ G and p ≤ q, then q ∈ G. • If ∆ ∈ W is dense in P, then G ∩ ∆ 6= ∅. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 6 jensen Subcomplete Forcing and L-Forcing If B = BA(P)W is the complete BA over P (as defined in W ), then it follows that G is P-generic over W iff F = FG = {b ∈ B | b ∩ G 6= ∅} is B-generic over W . Conversely, if F is B-generic, thus G = GF = {p | [p] ∈ F } is P-generic. We also note that if B is a complete BA, then hB \ {0}, ⊂i is a set of conditions, ∼ and there is an isomorphism σ : B ↔ BA(B \ {0}) defined by: σ(b) = {a | a ⊃ b}. Moreover, G is a B-generic filter iff it is a B \ {0}-generic set. When dealing with Boolean algebras, we shall often write: ”∆ is dense in B“ to mean ”∆ is dense in B \ {0}“. The Two Step Iteration Let A ⊆ B, where A, B are both complete. If (in some larger universe) G is A-generic W over V, then G′ = {b ∈ B | a ∈ G a ⊂ b} is a complete filter on B and we can form the factor algebra B/G′ (which we shall normally denote by B/G). It is not hard to see that B/G is then complete in V[G]. By the definition of the factor algebra there is a canonical homomorphism σ : B → B/G s.t. σ(b) ⊂ σ(c) ↔ ¬b ∪ c ∈ G′ . When the context permits we shall write b/G for σ(b). We now list some basic facts about this situation. Fact 1 Let B0 ⊆ B1 , B0 and B1 being complete. Let G0 be B0 -generic over V and let G̃ be B̃ = B1 /G-generic over V[G]. Set G1 = G0 ∗ G̃ =Df {b ∈ B1 | b/G0 ∈ G̃}. Then G1 is B1 -generic over V and V[G1 ] = V[G0 ][G̃]. Conversely we have: Fact 2 If G1 is B1 -generic over V and we set: G0 = B0 ∩ G1 , G̃ = {b/G0 | b ∈ G1 }. Then G0 is B0 -generic over V, G̃ is B1 /G0 -generic over V[G0 ] and G1 = G0 ∗ G̃. Fact 3 Let A ⊆ B and let h = hA,B as defined above. Then ◦ h(b) = [[b̌/G 6= 0]]A , ◦ G being the A-generic name. ◦ T T ([[ǎ ⊃ b̌]] ⇒ a) where a = [[ǎ ∈ G]] Proof. h(b) = {a ∈ A | a ⊃ b} = = T ◦ Fact 4 Let A ⊆ B and ◦ ◦ ◦ [[ǎ ⊃ b̌ → ǎ ∈ G]] = [[ a ∈ Ǎ(a ⊃ b̌ → a ∈ G)]] = [[b̌/G 6= 0]] QED(Fact 3) a∈A A a∈A V A ◦ b̌ ∈ B̌/G, where b̂ ∈ VA . There is a unique b ∈ B s.t. ◦ b = b̌/G. Proof. To see uniqueness, let ◦ ◦ ◦ b̌/G = b̌′ /G. Then ◦ b̌ \ b̌′ /G = 0. Hence h(b \ b′ ) = [[b̌ \ b̌′ /G 6= 0]] = 0. Hence b \ b′ = 0. Hence b ⊂ b′ . Similarly b′ ⊂ b. ◦ ◦ W To see the existence, note that ∆ = {a ∈ A | b a b = b̌/G} is dense in A. Let July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 7 Preliminaries ◦ ◦ ◦ b = b̌a /G for a ∈ X. Set: b = X be a maximal antichain in ∆. Let a S a ∩ ba . a∈X ◦ b = b̌/G, since if G is A-generic there is a ∈ X ∩ G by genericity. Hence Then ◦ b G = ba /G = b/G. QED(Fact 4) Fact 2 shows that, if B0 ⊆ B1 , then forcing with B1 is equivalent to a two step iteration: Forcing first by B0 to get V[G0 ] and then by a B̃ ∈ V[G0 ]. We now show the converse: Forcing by B0 and then by some B̃ is equivalent to forcing by a single B1 : Fact 5 Let B0 be complete and let ◦ ◦ ◦ B0 B is complete. There is B1 ⊇ B0 s.t. B0 ( B is isomorphic to B̌1 /G). (Hence, whenever G0 is B0 -generic, we have B1 /G0 ≃ ◦ B̃ =Df B G0 .) In order to prove this we first define: ◦ ◦ Definition Let A B is complete. B = A ∗ B is the BA defined as follows: Assume VA to be an identity model and set: |B| =Df {b ∈ VA | ◦ A b ∈ B}, b ⊂ c in B ←→Df A b ⊂ c. This defines B = h|B|, ⊂i. B is easily seen to be a BA with the operations: a ∩ b = that c s.t. A c = a ∩ b, a ∪ b = that c s.t. A c = a ∪ b, ¬b = that c s.t. A c = ¬b. ◦ Similarly, if hbi | i ∈ Ii is any sequence of elements of B, there is a B ∈ VA defined by: ◦ A ◦ B : Iˇ −→ B; ◦ A B(ǐ) = bi for i ∈ I. We then have: \ bi = that c s.t. A c= ◦ B(i), i∈Iˇ i∈I [ \ bi = that c s.t. A c= [ ◦ B(i), i∈Iˇ i∈I showing that B is complete. Now define σ : A → B by: σ(a) = that c s.t. ◦ A ◦ (a ∈ G ∧ c = 1) ∨ (a ∈ / G ∧ c = 0). σ is easily shown to be a complete embedding. Clearly, if G is A-generic, then σ ′′ G is σ ′′ A-generic, and V[G] = V[σ ′′ G]. Set G̃ = σ ′′ G, B̃ = B/G̃. We then have for b, c ∈ B: W b/G̃ ⊂ c/G̃ ←→ a ∈ G σ(a) ⊂ (¬b ∪ c) ←→ ←→ (¬bG ∪ cG ) = 1 ←− bG ⊂ cG (since σ(a)G = 1 for a ∈ G). July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 8 jensen Subcomplete Forcing and L-Forcing ∼ ◦ Hence there is k : B̃ ↔ B G defined by: k(b/G̃) = bG . Hence: ◦ ( B is isomorphic to B̌/G). ∼ If A = B0 and we pick B1 ,π : B ↔ B1 with πσ = id, then B1 satisfies Fact 5. QED A ◦ The algebra A ∗ B constructed above is often useful. General Iterations It is clear from the foregoing that an n-step iteration – i.e. the result of n successive generic extensions of V – can be adequately described by a sequence hBi | i < ni s.t. Bi ⊆ Bj for i ≤ j < n. The final model is the result of forcing with Bn−1 . What about transfinite iterations? At first glance it might seem that there is no such notion, but in fact we can define the notion by turning the previous analysis on its head. We define: Definition By an iteration of length α > 0 we mean a sequence hBi | i < αi of complete BA’s s.t. • Bi ⊆ Bj for i ≤ j < α. S Bi , i.e. there is no proper • If λ < α is a limit ordinal, then Bλ is generated by i<λ T S S Bi ⊂ B and X, X ∈ B for all X ⊂ B. B ⊂ Bλ s.t. i<λ If Gi is Bi+1 -generic and Gi = G ∩ Bi , then V[G] = V[Gi ][G̃i ] where G̃i = {b/Gi | b ∈ G} is B̃i = Bi+1 /Gi -generic. If G is λ-generic for a limit λ, then V[G] can be regarded as a ”limit“ of successive B̃i -generic extensions, where Gi = G ∩ Bi , B̃i = Bi+1 /Gi for i < λ. ◦ ◦ In practice, we usually at the i-th stage pick a B i s.t. BA), and arrange that: ◦ Bi Bi ( B i is a complete ◦ (B̌i /G is isomorphic to B). If the construction of the Bi ’s is sufficiently canonical, then the iteration is com◦ pletely characterized by the sequence of B i ’s. However, our definition of ”iteration“ gives us great leeway in choosing Bλ for limit λ < α. We shall make use of that freedom in these notes. Traditionally, however, a handfull of standard limiting procedures has been used. The direct limit takes Bλ as the minimal completion of the S Bi . It is characterized up to isomorphism by the property that Boolean algebra i<λ S S Bi \ {0}), there is then a unique Bi \ {0} lies dense in Bλ . (If B∗ = BA( i<λ S i<λ ∗ Bi \ {0}.) Another frequently isomorphism of Bλ onto B taking b to [b] for b ∈ i<λ used variant is the inverse limit , which can be defined as follows: By a thread in July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Preliminaries jensen 9 hBi | i < λi we mean a b = hbi | i < λi s.t. bj ∈ Bj \ {0} and hBi Bj (bj ) = bi for i ≤ j < λ. We call Bλ an inverse limit of hBi | i < λi iff T bi 6= 0 in Bλ . • If b is a thread, then b∗ = i<λ • The set of such b∗ is dense in Bλ . Bλ is then characterized up to isomorphism by these conditions. (If T is the set of V all threads, we can define a partial ordering of T by: b ≤ c iff i < λ bi ⊂ ci .) If we then set: B∗ = BA(T ), there is a unique isomorphism of Bλ onto B∗ taking b∗ to [b] for each thread b.) By the support of a thread we mean the set of j < λ s.t. bi 6= bj for all i < j. The countable support (CS) limit is defined like the inverse limit using only those threads which have a countable support. A CS iteration is one in which Bλ is a CS limit for all limit λ < α. (This is equivalent to taking the inverse limit at λ of cofinality ω and otherwise the direct limit.) Countable support iterations tend to work well if no cardinal has its cofinality changed to ω in the course of the iteration. Otherwise – e.g. if we are trying to iterate Namba forcing – we can use the revised countable support (RCS) iteration, which was invented by Shelah. The present definition is due to Donder: By an RCS thread we mean a thread b s.t. either there is i < λ s.t. bi Bi cf(λ̌) = ω or the support of b is bounded in λ. The RCS limit is then defined like the inverse limit, using only RCS threads. An RCS iteration is one which uses the RCS limit at all limit points. Note Almost all iterations which have been employed to date make use of sublimits of the inverse limit – i.e. {b∗ | b is a thread ∧ b∗ 6= 0} is dense in Bλ for all limit Q Bi )+ remains regular. In these notes, however, we shall λ. This means that ( i<λ see that it is sometimes necessary to employ larger limits which do not have this consequence. In dealing with iterations we shall employ the following conventions: If B = hBi | i < αi is an iteration we assume the VBi to be so constructed that VBi ⊆ VBj x ]]Bj for (in the sense of our earlier definition). In particular [[ϕ(~x )]]Bi = [[ϕ(~ x1 , . . . , xn ∈ VBi , i ≤ j, when ϕ is a Σ0 formula. We shall also often simplify the notation by using the indices i < α as in: hij for hBi Bj , i for Bi , [[ϕ]]i for [[ϕ]]Bj . If i0 < α and G is Bi0 -generic, we set: B/G = hBi0 +j /G | j < α − i0 i. We can assume the factor algebras to be so defined that Bi0 +h /G ⊆ Bi0 +j /G for S h ≤ j < α − i0 . (B̃ = Bi is a BA. Hence we can form B̃/G and identify Bi0 +j /G i<α with {b/G | b ∈ Bi0 +j }.) It then follows easily that B/G is an iteration in V[G]. July 21, 2012 15:2 10 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Chapter 1 Admissible sets 1.1 Introduction Let H = Hω be the collection of hereditarily finite sets. We use the usual Levy hierarchy of set theoretic formulae: Π0 = Σ0 = fmlae in which all quantifiers are bounded. W Σn+1 = fmlae x ϕ where ϕ is Πn . V Πn+1 = fmlae x ϕ where ϕ is Σn . The use of H offers an elegant way to develop ordinary recursion theory. Call a relation R ⊂ H n r.e. (or ”H-r.e.“) iff R is Σ1 -definable over H. We call R recursive (or H-recursive) iff it is ∆1 -definable (i.e. R and its complement ¬R are Σ1 -definable). Then R ⊂ ω n is rec (r.e.) in the usual sense iff it is the restriction of an H-rec. (H-r.e.) relation to ω. Moreover, there is an H-recursive function π : ω ↔ H s.t. R ⊂ H n is H-recursive iff {hx1 , . . . , xn i | R(π(x1 ), . . . , π(xn ))} is recursive. (Hence {hx, yi | π(x) ∈ π(y)} is recursive.) ⋆ ⋆ ⋆ ⋆ ⋆ This suggests a way of relativizing the concepts of recursion theory to transfinite domains: Let N = h|N |, ∈, A1 , A2 , . . .i be a transitive structure (with finitely or infinitely many predicates). We define: R ⊂ N n is N -r.e. (N -rec.) iff R is Σ1 (∆1 ) definable over N. Since N may contain infinite sets, we must also relativize the notion ”finite“: u is N -finite iff u ∈ N. There are, however, certain basic properties which we expect any recursion theory to possess. In particular: • If A is recursive and u finite, then A ∩ u is finite. • If u is finite and F : u → N is recursive, then F ′′ u is finite. The transitive structures N = h|N |, ∈, A1 , A2 , . . .i which yield a satisfactory recursion theory are called admissible. They were characterized by Kripke and Platek as those which satisfy the following axioms: 11 jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 12 Subcomplete Forcing and L-Forcing S (1) ∅, {x, y}, x are sets. (2) The Σ0 -axiom of subsets (Aussonderung) x ∩ {z | ϕ(z)} is a set, where ϕ is any Σ0 formula. (3) The Σ0 -axiom of collection V W V W V W x y ϕ(x, y) → u v x ∈ u y ∈ v ϕ(x, y) where ϕ is any Σ0 formula. Note Applying (3) to: x ∈ u → ϕ(x, y), we get: V W W V W x ∈ u y ϕ(x, y) −→ v x ∈ u y ∈ v ϕ(x, y). Note Kripke-Platek set theory (KP) consists of the above axioms together with the axiom of extensionality and the full axiom of foundation (i.e. for all formulae, not just Σ0 ones). These latter axioms of course hold trivially in transitive domains. KPC (KP with choice) is KP augmented by: Every set is enumerable by an ordinal. We now show that admissible structures satisfy the criteria stated above. Lemma 1 Let u ∈ M . Let A be ∆1 (M ). Then A ∩ u ∈ M . W W Proof. Let Ax ↔ y A0 yx, ¬Ax ↔ y A1 yx, where A0 , A1 are Σ0 . Then V W V W x y(A0 yx ∨ A1 yx). Hence there is v ∈ M s.t. x ∈ u y ∈ v(A0 yx ∨ A1 yx). W Hence u ∩ A = u ∩ {x | y ∈ v A0 yx} ∈ M . QED(Lemma 1) Before verifying the second criterion we prove: Lemma 2 M satisfies: V W W V W x ∈ u y1 . . . yn ϕ(x, ~y ) −→ v x ∈ u y1 . . . yn ∈ v ϕ(x, ~y ) for Σ0 formulas ϕ. V W Proof. Assume x ∈ u y1 . . . yn ϕ(x, ~y ). Then V W W x w(x ∈ u → y1 . . . yn ∈ w ϕ(x, ~y )). Hence there is v ′ ∈ M s.t. | {z } Σ0 V W W S x ∈ u w ∈ v ′ y1 . . . yn ∈ w ϕ(x, ~y ). Take v = v ′ . QED(Lemma 2) Finally we get: Lemma 3 Let u ∈ M , u ⊂ dom(F ), where F is Σ1 (M ). Then F ′′ u ∈ M . W V W Proof. Let y = F (x) ↔ z F ′ zyx, where F ′ is Σ0 (M ). Since x ∈ u y y = F (x), V W W W there is v s.t. x ∈ u y, z ∈ v F ′ zyx. Hence F ′′ u = v ∩ {y | x ∈ u z ∈ v F ′ zyx}. QED(Lemma 3) By similarly straightforward proofs we get: W Lemma 4 If Ry~x is Σ1 , so is y Ry~x. V V W Lemma 5 If Ry~x is Σ1 , so is y ∈ u Ry~x (since y ∈ u z ϕ(y, z) ↔ W V W v y ∈ v z ∈ v ϕ(y, z)). {z } | Σ0 Lemma 6 If R, Q ⊂ M n are Σ1 , then so are R ∪ Q, R ∩ Q. jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 13 Admissible sets Lemma 7 If R′ (y1 , . . . , yn ) is Σ1 and f (x1 , . . . , xm ) is a Σ1 function for i = 1, . . . , n, then R(f1 (~x ), . . . , fn (~x )) is Σ1 . Proof. R(f~ (~x )) ↔ W y1 . . . yn ( n V yi = fi (~x ) ∧ R(~y )). QED(Lemma 7) i=1 Note The boldface versions of Lemmas 4–7 follow immediately. Corollary 8 If the functions f (z1 , . . . , zn ), gi (~x ) (i = 1, . . . , n) are Σ1 in a parameter p, then so is h(~x ) ≃ f (g1 (~x ), . . . , gn (~x )). S Lemma 9 The following functions are ∆1 : x, x ∪ y, x ∩ y, x \ y (set difference), {x1 , . . . , xn }, hx1 , . . . , xn i, dom(x), rng(x), x′′ y, x ↾ y, x−1 , x × y, (x)ni , where: (hz0 , . . . , zn−1 i)i = zi ; (u)ni = ∅ otherwise;  x(z) if x is a function and z ∈ dom(x), x[z] = ∅ if not. Note As a corollary of Lemma 3 we have: If f is Σ1 , u ∈ M , u ⊂ dom(f ). Then f ↾ u ∈ M , since f ↾ u = g ′′ u, where g(x) ≃ hf (x), xi. Lemma 10 If f : M n+1 → M is Σ1 in the parameter p, then so are: F (u, ~x ) = {f (z, ~x ) | z ∈ u}, F ′ (u, ~x ) = hf (z, ~x ) | z ∈ ui. V W V W Proof. y = F (u, ~x ) ↔ z ∈ y v ∈ u z = f (y, ~z ) ∧ v ∈ u z ∈ y z = f (y, ~x ). But F ′ (u, ~x ) = {f ′ (z, ~x ) | z ∈ u}, where f ′ (y, ~x ) = hf (y, ~x ), ~x i. QED(Lemma 10) (Note The proof of Lemma 10 shows that, even if f is not defined everywhere, F is Σ1 in p, where: F (u, ~x ) ≃ {f (y, ~x ) | y ∈ u}, where this equation means that F (u, ~x ) is defined and has the displayed value iff f (y, ~x ) is defined for all y ∈ u. Similarly for F ′ .) Lemma 11 (Set Recursion Theorem) Let G be an n + 2-ary Σ1 function in the parameter p. Then there is F which is also Σ1 in p s.t. F (y, ~x ) ≃ G(y, ~x, hF (z, ~x ) | z ∈ yi) (where this equation means that F is defined with the displayed value iff F (z, ~x ) is defined for all z ∈ y and G is defined at hy, ~x, hF (z, ~x ) | z ∈ yii.) W Proof. Set u = F (y, ~x) ↔ f (ϕ(f, ~x ) ∧ hu, yi ∈ f ), where S ϕ(f, ~x ) ←→ (f is a function ∧ dom(f ) ⊂ dom(f ) ∧ V ∧ y ∈ dom(f ) f (y) = G(y, ~x, f ↾ y)). The equation is verified by ∈-induction on y. QED(Lemma 11) July 21, 2012 15:2 14 World Scientific Book - 9.75in x 6.5in jensen Subcomplete Forcing and L-Forcing Corollary 12 TC, rn are ∆1 functions, where TC(x) = the transitive closure of x = x ∪ [ TC(z), z∈x rn(x) = the rank of lub{rn(z) | z ∈ x}. Lemma 13 ω, On ∩ M are Σ0 classes. S V Proof. x ∈ On ↔ ( x ⊂ x ∧ z, w ∈ x(z ∈ w ∨ w ∈ z)), V x ∈ ω ↔ (x ∈ On ∧ ¬Lim(x) ∧ y ∈ x¬Lim(y)), where Lim(x) ↔ (x 6= 0 ∧ x ∈ S On ∧ x = x). Corollary 14 The ordinal functions α + 1, α + β, α · β, αβ , . . . are ∆1 . An even more useful version of Lemma 11 is Lemma 15 Let G be as in Lemma 11. Let h : M → M be Σ1 in p s.t. {hx, zi | x ∈ h(z)} is well founded. There is F which is Σ1 in p s.t., F (y, ~x ) ≃ G(y, ~x, hF (z, ~x ) | z ∈ h(y)i). The proof is just as before. We also note: Lemma 16.1 Let u ∈ Hω . Then the class u and the constant function f (x) = u are Σ0 . W V V Proof. ∈-induction on u: x ∈ u ↔ x = z, x = u ↔ ( z ∈ x z ∈ u ∧ z ∈ x). z∈u z∈u QED Lemma 16.2 If ω ∈ M , then the constant function x = ω is Σ0 . V V Proof. x = ω ↔ ( z ∈ x z ∈ ω ∧ ∅ ∈ x ∧ z ∈ x z ∪ {z} ∈ x). Lemma 16.3 If ω ∈ M , the constant for x = Hω is Σ1 (hence ∆1 ). V W W W S Proof. x = Hω ↔ ( z ∈ x u f n ∈ ω( n ⊂ u ∧ x ⊂ u ∧ f : n ↔ x)) ∧ ∅ ∈ V x ∧ z, w ∈ x({z, w}, z ∪ w ∈ x). Lemma 17 Fin, Pω (x) are ∆1 , where Fin = {x ∈ M | x < ω}, Pω (x) = Fin ∩ P(x). W W Proof. x ∈ Fin ↔ n ∈ ω f fin ↔ x, W V W W x∈ / Fin ↔ y(y = ω ∧ n ∈ y f n ⊂ x fin ↔ n), V V V y = Pω (x) ↔ u ∈ y(u ∈ Fin ∧ u ⊂ x) ∧ z ∈ x ({z} ∈ y ∧ u, v ∈ y u ∪ v ∈ y) QED The constructible hierarchy relative to a class A is defined by: L0 [A] = ∅; Lν+1 [A]d = Def(hLν [A], A ∩ Lν [A]i) [ Lλ [A] = Lν [A] for limit λ, ν<λ July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 15 Admissible sets where Def(A) is the set of B ⊂ A which are A-definable in parameters from A. We also define Lν = Lν [∅]. The constructible hierarchy over a set u is defined by: L0 (u) = TC({u}), Lν+1 (u) = Def(Lν (u)), [ Lλ (u) = Lν (u) for limit λ. ν<λ It is easily seen that: Lemma 18 If A ⊂ M is ∆1 (M ) in p, then hLν [A] | ν ∈ M i is ∆1 (M ) in p. • If u ∈ M , then hLν (u) | ν ∈ M i is ∆1 (M ) in u. By set recursion we can also define a sequence h<A ν | ν < ∞i s.t. • <A ν well orders Lν [A]. A • <A µ end extends <ν for ν ≤ µ. Then: Lemma 19 If A ∈ M is ∆1 (M ) in p, then h<A ν | ν ∈ M i is ∆1 (M ) in p. Definition LA ν = hLν [A], A ∩ Lν [A]i. hLA ν , B1 , B2 , . . .i = hLν [A], A ∩ Lν [A], B1 , B2 , . . .i. It follows easily that: Lemma 20 Let M = hLA α , B1 , . . .i be admissible. Then <M =Df S <A is a ν<α ∆1 (M ) well ordering of M . Moreover, there is a ∆1 (M ) map h : M → M s.t. h(x) = {z | z <M x}. Using this, it follows easily that every Σ1 (M ) relation is uniformizable by a Σ1 (M ) function. Thus the KP axioms give us a “reasonable” recursion theory. They do not suffice, however, to get Σ1 -uniformization. In fact, since we have not posited the axiom of choice, we do not even have N -finite uniformization. However, the admissible structures dealt with in these notes will almost always satisfy Σ1 -uniformization. This can happen in different ways. If N = LA τ =Df hLτ [A], Ai, there is a well ordering < of N s.t. the function h(x) = {z | z < x} is Σ1 . We can then uniformize W R(y, ~x ) as follows: Let R(y, ~x ) ↔ z R′ (y, z, ~x ), where R′ is Σ0 . R is then uniformized by: W V z(R′ (y, z, ~x ) ∧ hy ′ , z ′ i ∈ h(hy, zi)¬R(u′ , z ′~x )). The same holds for N = Lτ (a) where a is a transitive set with a well ordering constructible from a below τ . If N is a ZFC− model with a definable well ordering <, then every definable relation has a definable uniformization. If N ∗ = hN, A1 , A2 , . . .i is the result of adding all N -definable predicates to N , then the Σ1 (N ∗ ) relations are exactly the N -definable relations, so uniformization holds trivially. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 16 1.2 Subcomplete Forcing and L-Forcing Ill founded ZF− models We now prove a lemma about arbitrary (possibly ill founded) models of ZF− (where the language of ZF− may contain predicates other than ’∈’). Let A = hA, ∈A , B1 , B2 , . . .i be such a model. For X ⊂ A we of course write A|X = hX, ∈A ∩X 2 , . . .i. By the well founded core of A we mean the set of all x ∈ A s.t. ∈A ∩ C(x)2 is well founded, where C(x) is the closure of {x} under ∈A . Let wfc(A) denote the restriction of A to its well founded core. Then wfc(A) is a well founded structure satisfying the axiom of extensionality, and is, therefore, isomorphic to a transitive structure. Hence there is A′ s.t. A′ is isomorphic to A and wfc(A′ ) is transitive. We say that a model A of ZF− is solid iff wfc(A) is transitive and ∈wfc(A) =∈ ∩wfc(A)2 . Thus every consistent set of sentences in ZF− has a solid model. Note that if A is solid, then ω ⊂ wfc(A). By Σ0 -absoluteness we of course have: (1) wfc(A)  ϕ(~x ) ←→ A  ϕ(~x ) if x1 , . . . , xn ∈ wfc(A) and ϕ is a Σ0 -formula. By ∈-induction on x ∈ wfc(A) it follows that the rank function is absolute: (2) rn(x) = rnA (x) for x ∈ wfc(A). Using this we prove: Lemma 21 Let A be a solid model of ZF− . Then wfc(A) is admissible. Proof. Let ϕ be Σ0 and let V W (3) wfc(A))  x y ϕ(x, y, ~z ) where z1 , . . . , zn ∈ wfc(A). Let u ∈ wfc(A). By (3) and Σ0 absoluteness: V W (4) A  x ∈ u y ϕ(x, y, ~z ). Since A is a ZFC− model, there must then be v ∈ A of minimal A-rank rnA (v) s.t. V W (5) A  x ∈ u y ∈ v ϕ(x, y, ~z ). It suffices to note that rnA (v) ∈ wfc(A), hence rnA (v) = rn(v) and v ∈ wfc(A). (Otherwise there is r ∈ A s.t. A  r < rn(v) and there is v ′ ∈ A s.t. A  v ′ = {x ∈ v | rn(x) < r}. Hence v ′ satisfies (5) and rnA (v ′ ) < rnA (v). Contradiction!) By Σ0 absoluteness, then: V W (6) wfc(A)  x ∈ u y ∈ v ϕ(x, y, ~z ). QED (Lemma 21) As immediate corollaries we have: Corollary 21.1 Let δ = On ∩ wfc(A). Then Lδ (a) is admissible for a ∈ wfc(A). Corollary 21.2 LA δ = hLδ [A], A ∩ Lδ [A]i admissible whenever A is A-definable. jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 17 Admissible sets (Proof. We may suppose w.l.o.g. that A is one of the predicates of A.) Note In Lemma 21 we can replace ZF− by KP. In this form it is known as Ville’s Lemma. However, a form of Lemma 21 was first employed in our paper [NA] with Harvey Friedman. If memory serves us, the idea was due to Friedman. 1.3 Primitive Recursive Set Functions A function f : V → V is called primitive recursive successive applications of the following schemata: (i) (ii) (iii) (iv) (v) (pr) iff it is generated by f (~x ) = xi (here ~x is x1 , . . . , xn ) f (~x ) = {xi , xj } f (~x ) = xi \ xj f (~x ) = g(h1 (~x ), . . . , hm (~x )) S f (y, ~x ) = g(z, ~x ) z∈y (vi) f (y, ~x ) = g(y, ~x, hf (z, ~x ) | z ∈ yi) We call A ⊂ Vn a pr relation iff its characteristic function is a pr function. (However , a function can be a pr relation without being a pr function.) pr functions are ubiquitous. It is easily seen for instance that the functions listed in Lemma 9 are pr. Lemmas 4–7 hold with ’Σ1 ’ replaced by ’pr’. The functions TC(x), rn(x) are easily seen to be pr. We call f : Onn → V a pr function if it is the restriction of a pr function to On. The functions α + 1, α + β, α · β, αβ , . . . etc. are then pr. Since the pr functions are proper classes, the above discussion is carried out in second order set theory. However, all that needs to be said about pr functions can, in fact, be adequately expressed in ZFC. To do this we talk about pr definitions: By a pr definition we mean a finite list of schemata of the form (i)–(vi) s.t. • the function variable on the left side does not occur in a previous equation in the list. • every function variable on the right side occurs previously on the left side. Clearly, every pr definition s defines a pr function Fs . Moreover, for each s, Fs has a canonical Σ1 definition ϕs (y, x1 , . . . , xn ). (Indeed, the relation {hx, si | x ∈ Fs } is Σ1 .) The canonical definition has some remarkable absoluteness properties. If u is transitive, let Fsu be the function obtained by relativizing the canonical definition to u. Hence Fsu ⊂ Fs is a partial map on u. Then: • If u is pr closed, then Fsu = Fs ∩ u. • If α is closed under the functions ν + 1, ν · τ, ν τ , . . . etc., then Lα [A] is pr closed for every A ⊂ V. These facts are provable in ZFC− . The proofs can be found in [AS] or [PR] As corollaries we get: July 21, 2012 15:2 18 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing V[G] (1) Let V[G] be a generic extension of V. Then V ∩ Fs = FsV . − (2) Let A be a solid model of ZFC . Let A = wfc(A). Then FsA ∩ A = FsA = Fs . Proof. We prove (2). Clearly FsA = Fs , since A being admissible, is pr closed. But each x ∈ A is an element of a transitive pr closed u ∈ A, since A is admissible. Hence y = FsA (x) ↔ y = Fsu (x) ↔ y = FsA (x). QED jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Chapter 2 Barwise Theory Jon Barwise worked out the syntax and model theory of certain infinitary (but M finitary) languages on countable admissible structures M . In so doing, he created a powerful and flexible tool for set theorists, which enables us to construct transitive structures using elementary model theory. In this chapter we give an introduction to Barwise’ work, whose potential for set theory has, we feel, been unduly neglected. Let M be admissible. Barwise develops a first order theory in which arbitrary M -finite conjunctions and disjunctions are allowed. The predicates, however, have only a (genuinely) finite number of argument places and there are no infinite strings of quantifiers. If we wish to make use of the notion of M -finiteness, we must “arithmetize” the language – i.e. identify its symbols with objects in M . A typical arithmetization is: Predicates: Pxn = h0, hn, xii (x ∈ M , 1 ≤ n < ω) (Pxn = the x-th n-place predicate) Constants: cx = h1, xi (x ∈ M ) Variables: vx = h2, xi (x ∈ M ) Note The set of variables must be M -infinite, since otherwise a single formula could exhaust all the variables. We let P02 be the identity predicate (=) ˙ and 2 ˙ also reserve P1 as the ∈-predicate (∈), which will be a part of most interesting languages. By a primitive formula we mean P t1 . . . tn = h3, hP, t1 , . . . , tn ii, where P is an n-place predicate and t1 , . . . , tn are variables and constants. We then define: ¬ϕ = h4, ϕi, (ϕ ∨ ψ) = h5, hϕ, ψii, (ϕ ∧ ψ) = h6, hϕ, ψii, V (ϕ → ψ) = h7, hϕ, ψii, (ϕ ↔ ψ) = h8, hϕ, ψii, v ϕ = h9, hv, ϕii, W W W V V v ϕ = h10, hv, ϕii, and: f = h11, f i, f = h12, f i. The set Fml of 1-st order M -formulas is the smallest set X which contains all primitive formulae, is closed under ¬, ∨, ∧, →, ↔, and s.t. V W • If v is a variable and ϕ ∈ X, then v ϕ, v ϕ ∈ X. 19 jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 20 jensen Subcomplete Forcing and L-Forcing • If f = hϕi | i ∈ Ii ∈ M and ϕi ∈ X for i ∈ I, then V V V V ϕi =Df f are in I. W W i∈I ϕi =Df W W f and i∈I Then the usual syntactical notions are ∆1 , including: Fml, Cnst (set of constants), Vbl (set of variables), Sent (set of all sentences), Fr(ϕ) = the set of free variables in ϕ, and: ϕ(v1 , . . . , vn /t1 , . . . , tn ) ≃ the result of replacing all free occurences of the vbl vi by ti (where ti ∈ Vbl ∪ Const), as long as this can be done without any new occurence of a variable ti being bound; otherwise undefined. That Vbl, Const are ∆1 (in fact Σ0 ) is immediate. The characteristic function X of Fml is definable by a recursion of the form: X (x) = G(x, hX (z) | z ∈ TC(x)i). Similarly for the functions Fr(ϕ) and ϕ(~v /~). Then Sent = {ϕ | Fr(ϕ) = ∅}. t Note We of course employ the usual notation, writing ϕ(t1 , . . . , tn ) for ϕ(v1 , . . . , vn /t1 , . . . , tn ), where the sequence v1 , . . . , vn is taken as known. M -finite predicate logic has as axioms all instances of the usual predicate logical axiom schemata together with: _ _ ^ ^ ϕi for j ∈ u ∈ M. ϕi −→ ϕj , ϕj −→ i∈u i∈u The rules of inference are: ϕ, ϕ → ψ (modus ponens), ψ ψ→ϕ ϕ→ψ V , W for x ∈ / Fr(ϕ), ϕ→ xψ xψ→ϕ ψi → ϕ (i ∈ u) ϕ → ψi (i ∈ u) V V W W , . ϕ→ ϕi ψi → ϕ i∈u i∈u We say that ϕ is provable from a set of statements A if ϕ is in the smallest set which contains A and the axioms and is closed under the rules of inference. We W W write A ⊢ ϕ to mean that ϕ is provable from A. (Note: By the last rule, ∅ → ϕ W W V V for every ϕ, hence ⊢ ¬ ∅. Similarly ⊢ ∅.) A formula is provable if and only if it has a proof. Because we have not assumed choice to hold in our admissible structure M , we must use a somewhat unorthodox concept of proof, however. Definition By a proof from A we mean a sequence hpi | i < αi s.t. α ∈ On and for each i < α, if ψ ∈ pi , then either ψ ∈ A or ψ is an axiom or ψ follows from S ph by a single application of one of the rules. h<i S p = hpi | i < αi is a proof of ϕ iff ϕ ∈ pi . i<α July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 21 Barwise Theory If A is Σ1 (M ) in a parameter q it follows easily that {p ∈ M | p is a proof from A} is Σ1 (M ) in the same parameter. It is also easily seen that A ⊢ ϕ iff there exists a proof of ϕ from A. A more interesting conclusion is: Lemma 1 Let A be Σ1 (M ). Then A ⊢ ϕ iff there is an M -finite proof of ϕ from A. Proof . (←) is trivial. We prove (→). Let X = the set of ϕ s.t. there exists a p ∈ M which proves ϕ from A. Claim {ϕ | A ⊢ ϕ} ⊂ X. Proof . We know that A ⊂ X and all axioms lie in X. Hence it suffices to show that X is closed under the rules of proof. This must be demonstrated rule by rule. As an example we show: V V Claim Let ϕ → ψi ∈ X for i ∈ u, where u ∈ M . Then ϕ → ψi ∈ X. i∈u Proof . Let P (p, ψ) mean: p is a proof of ψ from A. Then P is Σ1 (M ). By our assumption: V W (1) i∈u p P (p, ϕ → ψi ). W Now let P (p, ψ) ↔ z P ′ (z, p, ψ), where P ′ is Σ0 . We then have: V W W (2) i∈u z p P ′ (z, p, ϕ → ψi ) whence follows easily that there is v ∈ M with: V W W (3) i∈u z∈v p ∈ v P ′ (z, p, ϕ → ψi ). W W Set w = {p ∈ v | i ∈ u z ∈ v P ′ (z, p, ψ)}. Then V W (4) i∈u p ∈ w P (p, ϕ → ψi ) and w consists of proofs from A. Let α ∈ M , α ≥ dom(p) for all p ∈ w. Define a proof p∗ of length α + 1 by: S  {pi | p ∈ w ∧ i ∈ dom(p)} for i < α, V p∗ (i) = {ϕ → V ψi } for i = α.  i∈u ∗ Then p ∈ M proves ϕ → V V ψi from A. QED(Lemma 1) i∈u From this we get the M -finiteness lemma: Lemma 2 Let A be Σ1 (M ). Then A ⊢ ϕ iff there is u ∈ M s.t. u ⊂ A and u ⊢ ϕ. Proof . (←) is trivial. We prove (→). Let p ∈ M be a proof of ϕ from A. Let u = the set of ψ s.t. for some i ∈ dom(p), S ph by a single application ψ ∈ pi , but ψ is not an axiom and does not follow from h<i of a rule. Then u ∈ M , u ⊂ A, and p is a proof from u. Hence u ⊢ ϕ.QED(Lemma 2) Another consequence of Lemma 1 is July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 22 Subcomplete Forcing and L-Forcing Lemma 3 Let A be Σ1 (M ) in q. Then {ϕ | A ⊢ ϕ} is Σ1 (M ) in the same parameter q (uniformly in the Σ1 definition of A from q). W Proof . {ϕ | A ⊢ ϕ} = {ϕ | p ∈ M p proves ϕ from A}. QED Corollary 4 Let A be Σ1 (M ) in q. Then “A is consistent” is Π1 (M ) in the same parameter q (uniformly in the Σ1 definition of A from q). Note that, since u ∈ M is uniformly Σ1 (M ) in itself, we have: Corollary 5 {hu, ϕi | u ∈ M ∧ u ⊢ ϕ} is Σ1 (M ). Similarly: Corollary 6 {u ∈ M | u is consistent} is Π1 (M ). Note Call a proof p strict iff pi = 1 for i ∈ dom(p). This corresponds to the more usual notion of proof. If M satisfies the axiom of choice in the form: Every set is enumerable by an ordinal, then Lemma 1 holds with “strict proof” in place of “proof”. We leave this to the reader. Languages We will normally not employ all of the predicates and constants in our M -finitary first order logic, but cut down to a smaller set of symbols which we intend to interpret in a model. Thus we define a language to be a set L of predicates and constants. By a model of L we mean a structure A = h|A|, htA | t ∈ Lii s.t. |A| 6= ∅, P A ⊂ |A|n whenever P is an n-place predicate, and cA ∈ |A| whenever |A| is a constant. By a variable assignment we mean a map f : Vbl → A (Vbl being the set of all variables). The satisfaction relation A  ϕ[f ] is defined in the usual way, where A  ϕ[f ] means that the formula ϕ becomes true in A if the free variables in ϕ are interpreted by f . We leave the definition to the reader, remarking only that: ^ ^ V ϕi [f ] iff i ∈ u A  ϕi [f ], A i∈u A _ _ i∈u ϕi [f ] iff W i ∈ u A  ϕi [f ]. We adopt the ususal conventions of model theory, writing A = h|A|, tA 1 , . . .i if we think of the predicates and constants of L as being arranged in a fixed sequence t1 , t2 , . . . Similarly, if ϕ = ϕ(v1 , . . . , vn ) is a formula in which at most the variables v1 , . . . , vn occur free, we write: A  ϕ[x1 , . . . , xn ] for: A  ϕ[f ] where f (vi ) = xi (i = 1, . . . , n). If ϕ is a statement, we write: A  ϕ. If A is a set of statements we write: A  A to mean: A  ϕ for all ϕ ∈ A. jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 23 Barwise Theory The correctness theorem says that if A is a set of L-statements and A  A, then A is consistent. (We leave this to the reader.) Barwise’ Completeness Theorem says that the converse holds if our admissible structure M is countable: Theorem 7 Let M be a countable admissible structure. Let A be a set of statements in the M -language L. If A is consistent in M -finite predicate logic, then A has a model A. Proof (sketch). We make use of the following theorem of Rasiowa and Sikorski: Let S B be a Boolean algebra. Let Xi ⊂ B (i < ω) s.t. the Boolean union Xi = bi exists in the sense of B. Then B has an ultrafilter U s.t. bi ∈ U ←→ Xi ∩ U 6= ∅ for i < ω. (Proof . Successively choose ci (i < ω) by c0 = 1, ci+1 = ci ∩ b 6= 0, where b ∈ Xi ∪ {¬bi }. Let U = {a ∈ B | Vi ci ⊂ a}. Then U is a filter and extends to an ultrafilter on B.) Extend the language L by adding an M -infinite set C of new constants. Call the extended language L∗ and set: [ϕ] = {ψ | A ⊢ ψ ↔ ϕ} for L∗ -statements ϕ. Then B = {[ϕ] | ϕ ∈ StL∗ } in the Lindenbaum algebra of L∗ with the operations: [ϕ] ∪ [ψ] = [ϕ ∨ ψ], [ϕ] ∩ [ψ] = [ϕ ∧ ψ], ¬[ϕ] = [¬ϕ], h^ h_ ^ i \ _ i [ (u ∈ M ), ϕi [ϕi ] = (u ∈ M ), ϕi [ϕi ] = i∈u [ c∈C i∈u i∈u W [ϕ(c)] = [ v ϕ(v)], \ c∈C i∈u V [ϕ(c)] = [ v ϕ(v)]. The last two equations hold because the constants in C, which do not occur in the axioms A, behave like free variables. By Rasiowa and Sikorski there is then an ultrafilter U on B which respects the above operations. We define a model A = h|A|, htA | t ∈ Lii as follows: For c ∈ C set [c] = {c′ ∈ C | [c = c′ ] ∈ U }. If P ∈ L is an n-place predicate, set: P A ([c1 ], . . . , [cn ]) ←→ [P c1 . . . cn ] ∈ U. If t ∈ L is a constant set: tA = [c], where c ∈ C, [t = c] ∈ U. A straighforward induction then shows: A  ϕ[[c1 ], . . . , [cn ]] ←→ [ϕ(c1 , . . . , cn )] ∈ U July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 24 Subcomplete Forcing and L-Forcing for formulae ϕ = ϕ(v1 , . . . , vn ) with at most the free variables v1 , . . . , vn . In particular A  ϕ ↔ [ϕ] ∈ U for L∗ -statements ϕ. Hence A  A, since [ϕ] = 1 for all ϕ ∈ A. QED(Theorem 7) Combining the completeness theorem with the M -finiteness lemma, we get the well known Barwise compactness theorem: Corollary 8 Let M be countable. Let L be ∆1 and A be Σ1 . If every M -finite subset of A has a model, then so does A. By a theory or axiomatized language we mean a pair L = hL0 , Ai s.t. L0 is a language and A a set of L0 -statements. We say that A models L iff A is a model of L0 and A  A. We also write: L ⊢ ϕ for (A ⊢ ϕ ∧ ϕ ∈ FmlL0 ). We say that L = hL0 , Ai is Σ1 (M ) (in the parameter p) iff L0 is ∆1 (M ) (in p) and A is Σ1 (M ) (in p). Similarly for: L is ∆1 (M ) (in p). ⋆⋆⋆⋆⋆ ˙ We now consider the class of axiomatized languages containing a fixed predicate ∈, the special constants x (x ∈ M ) (We can set e.g. x = h1, h0, xii.) and the basic axioms • Extensionality W W V v = z) (x ∈ M ) • v(v ∈˙ x ↔ z∈x (Further predicates, constants, and axioms are allowed, of course.) We call any such theory an “∈ -theory”. Then: Lemma 9 Let A be a solid model of the ∈-theory L. Then xA = x ∈ wfc(A) for x ∈ M. Proof . ∈-induction on x. Definition Let L be an ∈-theory. ZF− L is the set of (really) finite L-statements which are axioms of L. (Similarly for ZFC− L .) − − We write L ⊢ ZF for L ⊢ ZFL . (Similarly for L ⊢ ZFC− .) ⋆⋆⋆⋆⋆ ∈-theories are a suseful tool in set theory. We now bring some typical applications. We recall that an ordinal α is called admissible if Lα is admissible and admissible in a ⊂ α if Laα = hLα [a], ai is admissible. Lemma 10 Let α > ω be a countable admissible ordinal. There is a ⊂ ω s.t. α is the least ordinal admissible in a. This follows straightforwardly from: Lemma 11 Let M be a countable admissible structure. Let L be a consistent Σ1 (M ) ∈-theory s.t. L ⊢ ZF− . Then L has a solid model A s.t. On ∩ wfc(A) = On ∩ M . jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 25 Barwise Theory We first show that Lemma 11 implies Lemma 10, and then prove Lemma 11. Take M = Lα , where α is as in Lemma 10. Let L be the M -theory with: Predicate: ∈˙ ◦ Constants: x (x ∈ M ), a ◦ Axioms: Basic axioms + ZF− , and β is not admissible in a (β < α). Then L is consistent, since hHω1 , ∈, ai is a model, where a is any a ⊂ ω which codes a well ordering of type ≥ α (and x is interpretedly x for x ∈ M ). Now let A be a solid model of L s.t. On ∩ wfc(A) = α. Then wfc(A) is admissible by ◦ Chapter 1, Lemma 21. Hence so is Laα , where a = a A . But β is not admissible in a for ω < β < α, since “β is admissible in a” is Σ1 (Laα ); hence the same Σ1 statement would hold of β in A. Contradiction! QED(Lemma 10) Note Pursuing this method a bit further we can prove: Let ω < α0 < . . . < αn−1 be a sequence of countable admissible ordinals. There is a ⊂ ω s.t. αi = the i-th α > ω which is admissible in a (i < n). A similar theorem holds for countable infinite sequences, but the proof requires forcing and is much more complex. It is given in §5 and §6 [AS] We now prove Lemma 11 by modifying the proof of the completeness theorem. Let Γ(v) be the set of formulae v ∈ On, v > β (β ∈ M ). Add an M -infinite (but ∆1 (M )) set E of new constants to L. Let L′ be L with the new constants and the new axioms Γ(e) (e ∈ E). Then L′ is consistent, since any M -finite subset of the axioms can be modeled by interpreting the new constants as ordinals in wfc(A), A being any solid model of L. As in the proof of completeness we then add a new class C of constants which is not M -finite. We assume, however, that C is ∆1 (M ). We add no further axioms, so the elements of C behave like free variables. The so extended language L′′ is clearly Σ1 (M ). Now set: [ [ {e < v}. ∆(v) = {v ∈ / On} ∪ {v ≤ β} ∪ β∈M e∈E S Claim Let c ∈ C. Then {[ϕ] | ϕ ∈ ∆(c)} = 1 in the Lindenbaum algebra of L′′ . Proof . Suppose not. Set ∆′ = {¬ϕ | ϕ ∈ ∆(c}. Then there is an L′′ statement ψ s.t. A ∪ {ψ} is consistent, where L′′ = hL′′0 , Ai and A ∪ {ψ} ⊢ ∆′ . Pick an e ∈ E which does not occur in ψ. Let A∗ be the result of omitting the axioms Γ(e) from A. Then A∗ ∪ {ψ} ∪ Γ(e) ⊢ c ≤ e. By the M -finiteness lemma there is β ∈ M s.t. A∗ ∪ {ψ} ∪ {β ≤ e} ⊢ c ≤ e. But e behaves here like a free variable, so A∗ ∪ {ψ} ⊢ c ≤ β. But A ⊃ A∗ and A ∪ {ψ} ⊢ β < c. Thus A ∪ {ψ} is inconsistent. Contradiction! QED(Claim) Now let U be an ultrafilter on the Lindenbaum algebra of L′′ which respects both the S operations listed in the proof of the completeness theorem and the unions {[ϕ] | ϕ ∈ ∆(c)} for c ∈ C. Let X = {ϕ | [ϕ] ∈ U }. Then as before, L′′ has a model A, all of whose elements have the form cA for a c ∈ C and such that A  ϕ ↔ ϕ ∈ X for L′′ -statements ϕ. We assume w.l.o.g. that A is solid. It suffices to show that July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 26 jensen Subcomplete Forcing and L-Forcing Y = {x ∈ A | x > ν in A for all v ∈ m} has no minimal element in A. Let x ∈ Y , x = cA . Then A  e < c for some e ∈ E. But eA ∈ Y . QED(Lemma 11) Another – very typical – application is: Lemma 12 Let W be an inner model of ZFC. Suppose that, in W , U is a normal measure on κ. Let τ > κ be regular in W and set M = hHτW , U i. Assume that M is countable in V. Then for any α ≤ κ there is M = hH, U i s.t. U is a normal measure in M and M iterates to M in exactly α many steps. (Hence M is iterable, since M is). Proof . The case α = 0 is trivial, so assume α > 0. Let δ be least s.t. Lδ (M ) is admissible. Then N = Lδ (M ) is countable. Let L be the ∈-theory on N with: Predicate: ∈˙ ◦ Constants: x (x ∈ N ), M ◦ ◦ ◦ ◦ Axioms: The basic axioms; ZFC− ; M = hH, U i is a transitive ZFC− model; M iterates to M in α many steps. It suffices to prove: Claim L is consistent. We first show that the claim implies the theorem. Let A be a solid model of L. ◦ Then N ⊂ wfc(A). Hence M, M ∈ wfc(A), where M = M A . There is hM i | i < αi which, in A, is an iteration from M to M . But then hM i | i < αi ∈ wfc(A) really is an iteration by absoluteness. QED We now prove the claim. Case 1 α < κ. Iterate hW, U i α many times, getting hWi , Ui i (i ≤ α) with iteration maps πij : hWi , Ui i ≺ hWj , Uj i. Set Mi = π0i (M ). Then hMi , Ui i (i ≤ α) is the iteration ′ of hM, U i with maps πij = πij ↾ Mi . It suffices to show that Lα = π0,α (L) is consistent. This is clear, however, since hHτ + , M i models Lα (with M interpreting ◦ ◦ the constant M α = π0,α (M )). QED(Case 1) Case 2 α = κ. This time we iterate hW, U i β many times where π0β (κ) = β and β ≤ κ+ . hHτ + , M i again models Lβ . QED(Lemma 12) Barwise theory is useful in situations where one is given a transitive structure Q and wishes to find a transitive structure Q with similar properties inside an inner model. Another tool used in such situations is Schoenfield’s lemma, which, however requires coding Q by a real. Unsurprisingly, Schoenfield’s lemma can itself be derived from Barwise theory. We first note the well known fact that every Σ12 condition on a real is equivalent to a Σ1 (Hω1 ) condition, and conversely. Thus it suffices to show: Lemma 13 Let Hω1  ϕ[a], a ⊂ ω, where ϕ is Σ1 . Then  ϕ[a]. HωL[a] 1 July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Barwise Theory jensen 27 W Proof . Let ϕ = z ψ, where ψ is Σ0 . Let Hω1  ψ[z, a], where rn(z) < α and α is admissible in a. Let L be the language on Lα (a) with: Predicate: ∈˙ ◦ Constants: z , x (x ∈ Lα (a)) ◦ Axioms: Basic axioms, ZFC− , ψ( z , a). Then L is consistent since hHω1 , zi is a model. Applying Löwenheim-Skolem in ◦ ◦ L(a), we find a countable α and a map π : Lα (a) ≺ Lα (a). Let w.l.o.g. π( z ) = z and let L be defined over Lα (a) like L over Lα (a). Then L is consistent and has a ◦ solid model A in L(a). Let z = z A . Then z ∈ L(a) and Hω1  ψ[z, a] in L(a). QED(Lemma 13) July 21, 2012 15:2 28 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Chapter 3 Subcomplete Forcing 3.1 Introduction In §10 of [PF] Shelah defines the notion of complete forcing: Definition Let B be a complete BA. B is a complete forcing iff for sufficiently large θ we have: Let B ∈ Hθ . Let σ : H ≺ H, where H is countable and transitive. Let σ(B) = B. If G is B-generic over H, then there is b ∈ B which forces that, that whenever G ∋ b is B-generic, then σ ′′ G ⊂ G. Note If G, G, H, H, σ are as above, then σ extends uniquely to a σ ∗ s.t. σ ∗ : H[G] ≺ H[G] and σ ∗ (G) = G. Proof . To see uniqueness, note that each x ∈ H[G] has the form x = tG where t ∈ H is a B-name. Thus σ ∗ (x) = σ(t)G . To see existence, note that: W G H[G]  ϕ(tG b∈Gb H ϕ(t1 , . . . , tn ) −→ 1 , . . . , tn ) ←→ B W H −→ b ∈ G B ϕ(σ(t1 ), . . . , σ(tn )) −→ H[G]  ϕ(σ(t1 )G , . . . , σ(tn )G ). Hence there is σ ∗ : H[G] ≺ H[G] defined by: σ ∗ (tG ) = σ(t)G . But then σ ∗ ⊃ σ since ˇ G = σ(x) σ ∗ (x) = σ ∗ (x̌G ) = σ(x) ˙ be the B-generic name and Ġ the B-generic name we then for x ∈ H. Letting G have: ˙ G ) = ĠG = G. σ ∗ (G) = σ ∗ (G QED Lemma 1.1 Let B be a complete forcing. Let G be B-generic. Then V[G] has no new countable sets of ordinals. Proof . Let f : ω̌ → On. Claim f G ∈ V. Suppose not. Then b f ∈ / V̌ for some b. Let θ be big enough and let σ : H ≺ Hθ s.t. σ(f , b, B) = f, b, B, where H is countable and transitive. Let G ∋ b be B-generic over H. Let G be B-generic s.t. σ ′′ G ⊂ G. Let σ ∗ be the above mentioned extension 29 jensen July 21, 2012 15:2 30 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing of σ. Then σ ∗ (f G ) = f G . But clearly σ ∗ (f G ) = σ ′′ f G ∈ V, where b = σ(b) ∈ G. Contradiction! QED(Lemma 1) We note without proof that Lemma 1.2 If B is the result of a countable support iteration of complete forcings, then B is complete. Remark In fact, the notion of complete forcing reduces to that of an ω-closed set of conditions. (P is called ω-closed iff whenever hpi | i < ωi is a sequence with pi ≤ pj for all j ≤ i, then there is q with q ≤ pi for all i.) It is shown in [FA] that: Lemma 1.3 B is a complete forcing iff it is isomorphic to BA(P) for some ω-closed set of conditions P. The properties of ω-closed forcing are well known and Lemmas 1.1, 1.2 follow easily from Lemma 1.3. The knowledgable reader will recognize the complete forcings as being a subclass of Shelah’s proper forcings. Proper forcings satisfy Lemma 1.2 but not Lemma 1.1. In fact, many proper forcings add new reals. However, a proper forcing can never change the cofinality of an uncountable regular cardinal to ω. Thus, the notion is useless in dealing e.g. with Namba forcing. What we want is a class of forcings which do not add new reals but do permit new sets of ordinals – even to the extent of changing cofinalities. We of course want these forcings to be iterable – i.e. some reasonable analogue of Lemma 1.2 should hold. The proof of Lemma 1.1 gives us a clue as to how such a class might be defined: The proof depends strongly on the fact that σ ′′ G ⊂ G for a σ ∈ V. Instead, we might require that, if H, σ, θ, B, G are as in the definition of “completed forcing”, then there is b ∈ B which forces that, if G ∋ b is B-generic, there is σ ′ ∈ V[G] s.t. σ ′ : H ≺ Hθ , σ ′ (B) = B and σ ′ ′′ G ⊆ G. We can even require b to force σ ′ (s) = σ(s) for an arbitrarily chosen s ∈ H. If we now try to carry out the proof of Lemma 1 with a σ ′ : H ≺ Hθ s.t. σ ′ (f , b, B) = f, b, B, in place of σ, we can conclude only that f G = σ ′ ′′ f G . Since we do not know that σ ′ ∈ V, we cannot conclude that f G ∈ V. However, if we assume f : ω → ω, then f G = σ ′ ′′ f G , where f G ∈ V and f G ⊂ ω 2 . Since σ ′ ↾ ω = id, we can then conclude that f G ∈ V. Thus such forcings will add no reals, but may permit us to add new countable sets of ordinals. In order to carry out this program we must address several difficulties, the first being this: Suppose that Hθ has definable Skolem functions. (This is certainly the case if V = L.) We could then form σ : H ≺ Hθ s.t. σ(b, f , B) = b, f, B simply by transitivizing the Skolem closure of {b, f, B}. But then σ is the only possible elementary map to Hθ with σ(b, f , B) = b, f, B. Thus we perforce have: σ ′ = σ. In order to avoid this we must place a stronger condition on H which implies the possibility of many maps to the top. We shall define such a condition for the case that H is a ZFC− -model. jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing jensen 31 Definition Let N be transitive. N is full iff ω ∈ N and there is γ s.t. Lγ (N ) models ZFC− and N is regular in Lγ (N ) – i.e. if f : x → N , x ∈ N , f ∈ Lγ (N ), then rng(f ) ∈ N . It follows that N itself is a ZFC− model. In fact, regularity in Lγ (N ) is equivalent to saying that N models 2nd order ZFC− in Lγ (N ). If N is full and σ : N ≺ N , then there will, indeed, be many different maps σ ′ : N ≺ N . Since fullness is a property of ZFC− models, however, we shall have to reformulate Shelah’s definition so that we do not work directly with Hθ but rather with ZFC− models containing Hθ . It also turns out that, in order to prove iterability, we must apparently impose a stronger similarity between σ ′ and σ than we have hitherto stated. In order to formulate this we define: Definition Let B be a complete BA. δ(B) = the smallest cardinality of a set which lies dense in B \ {0}. Note If we were working with sets P of conditions rather than complete BA’s, we would normally choose P to have cardinality δ(BA(P)). Hence the above definition would be superfluous and we would work with P instead. − Definition Let N = LA model. Let τ =Df hLτ [A], ∈, A ∩ Lτ [A]i be a ZFC X ∪ {δ} ⊂ N . CδN (X) =Df the smallest Y ≺ N s.t. X ∪ δ ⊂ Y. We are now ready to define: Definition Let B be a complete BA. B is a subcomplete forcing iff for sufficiently large cardinals θ we have: B ∈ Hθ and for any ZFC− model N = LA τ s.t. θ < τ and Hθ ⊂ N we have: Let δ : N ≺ N where N is countable and full. Let σ(θ, s, B) = θ, s, B where s ∈ N . Let G be B-generic over N . Then there is b ∈ B \ {0} s.t. whenever G ∋ b is B-generic over V, there is σ ′ ∈ V[G] s.t. (a) (b) (c) (d) σ′ : N ≺ N , σ ′ (θ, s, B) = θ, s, B, CδN (rng(σ ′ )) = CδN (rng(σ)) where δ = δ(B), σ ′ ′′ G ⊂ G. (Hence σ ′ extends uniquely to a σ ∗ : N [G] ≺ N [G] s.t. σ ∗ (G) = G.) Note We define N [G] in such a way that A is still a predicate. Thus N = LA τ is N [G]-definable. Note This is expressible in V, since the last part can be expressed as: ◦ W b∈B b G being the generic name. ◦ ˇ , Ň , σ◦, G, G), ϕ(B̌, N July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 32 Subcomplete Forcing and L-Forcing Note If we omitted (c) from the definition of subcompleteness, the resulting class of forcings would still satisfy Lemma 1.2 for countable support iterations of length ≤ ω2 . Since such forcings might change the cofinality of ω2 to ω, we would thereafter have to use the revised countable support (RCS) iteration. (We will also have to make some further assumptions on the component forcings Bi of the iteration hBi | i < αi.) (c) appears to be needed to get past regular limits points λ of the iteration s.t. λ > δ(Bi ) for i < λ. Definition θ verifies the subcompleteness of B iff θ is as in the definition of subcompleteness. In the following discussion write ’ver(B, θ)’ to mean ’θ verifies the subcompleteness of B’. Now let B ∈ Hθ and let θ′ > H θ be a cardinal. A Löwenheim-Skolem argument that, in order to determine whether ver(B, θ), we need only consider V[G] for B-generic N = LA τ s.t. N ∈ Hθ ′ . By the well known fact: Hθ ′ [G] = (Hθ ′ ) G, where B ∈ Hθ′ , we see that, in fact, the definition of ver(B, θ) relativizes to Hθ′ – i.e. Lemma 2.1 Let B ∈ Hθ . Let θ′ > H θ be a cardinal. The statement ver(B, θ) is absolute in Hθ′ . This holds in particular for θ′ = (H θ )+ . But then the elements of Hθ′ can be coded by subsets of Hθ and we get: Lemma 2.2 Let θ > ω1 be a cardinal. {B | ver(B, θ)} is uniformly 2nd order definable over Hθ . Hence: Corollary 2.3 absolute in W . Let W be an inner model s.t. P(Hθ ) ⊂ W . Then ver(B, θ) is Finally, we note: Lemma 2.4 Let θ verify the subcompleteness of B. Then B is subcomplete. (Thus “sufficiently large θ” can be replaced by “some θ” in the definition of ’subcomplete’.) Proof of Lemma 2.4. It suffices to show: Claim Let B ∈ Hθ . Let θ′ > H θ be a cardinal. Then ver(B, θ′ ). Proof . We can assume w.l.o.g. that θ is least with ver(B, θ). Then by Lemma 2.1: (1) Hθ′  θ is least s.t. ver(B, θ). ′ Now let N = LA τ s.t. θ < τ and Hθ ′ ⊂ N . Then θ < τ and Hθ ⊂ N . Let σ : N ≺ N where N is countable and full. Let σ(B, θ′ , s) = B, θ′ , s. By (1) there is θ s.t. σ(θ) = θ. By ver(B, θ) there is then a b ∈ B with the desired property. QED(Lemma 2.4) jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing jensen 33 When actually verifying the subcompleteness of a specific B we often find it convenient to employ an additional parameter. Thus we define: Definition hθ, pi verifies the subcompleteness of B (ver(B, θ, p)) iff p, B ∈ Hθ and for any ZFC− model N = LA τ with θ < τ and Hθ ⊂ N we have: Let σ : N ≺ N where N is countable and full. Let σ(p, θ, s, B) = p, θ, s, B. Let G be B-generic over N . Then the previous conclusion holds. The natural analogues of Lemma 2.1 – Corollary 2.3 follow as before. But then we can repeat the proof of Lemma 2.4 to get: Lemma 2.5 Let hθ, pi verify the subcompleteness of B. Then B is subcomplete. This will often be tacitly used in verifications of subcompleteness. 3.2 Liftups In order to better elucidate the concept of fullness, we make a digression on the topic of cofinal embeddings. Definition Let A, A be models which satisfy the extensionality axiom. Let π : A → A be a structure preserving map. We call π cofinal (in symbols: π : A → A cofinally) iff for all x ∈ A there is u ∈ A s.t. x ∈A π(u). Note In this definition we did not require A, A to be transitive or even well founded. Most of our applications will be to transitive models, but we must occasionally deal with ill founded structures. We shall, however, normally assume such structures to be solid in the sense of Chapter 1. (I.e. the well founded core of A (wfc(A)) is transitive and ∈A ∩wfc(A)2 =∈ ∩wfc(A)2 .) Definition Let τ be a cardinal in A. HτA = the set of x s.t. A  x ∈ Hτ . Note Even if A were a transitive ZFC− model, we would not necessarily have: HτA ∈ A. Definition Let τ ∈ A be a cardinal in A. We call π : A → A τ -cofinal iff for all x ∈ A there is u ∈ A s.t. u < τ in A and x ∈A π(u). We shall generally work with elementary embeddings but must sometimes consider a finer degree of preservation: Definition π : A → A is Σn -preserving (π : A →Σn A) iff for all Σn -formulae ϕ and all x1 , . . . , xn ∈ A: A  ϕ[x1 , . . . , xn ] ←→ A  ϕ[π(x1 ), . . . , π(xn )]. Definition Let A be a solid model of ZFC− . Let τ ∈ wfc(A) be an uncountable cardinal in A. Set H = HτA . (Hence H ⊂ wfc(A).) Let π : H →Σ0 H cofinally, where H is transitive. Then by a liftup of hA, πi we mean a pair hA, πi s.t π ⊃ π, July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 34 Subcomplete Forcing and L-Forcing H ⊂ wfc(A), and π : A →Σ0 A τ -cofinally, where A is solid. (We also say: π : A → A is a liftup of A by π : H → H.) Lemma 3.1 Let A, τ , H, H, π be as in the above definition. The liftup hA, πi of hA, πi (if it exists) is determined up to isomorphism (i.e. if hA′ , π ′ i is another ∼ liftup, there is σ : A ↔ A′ with σπ = π ′ ). Proof . Set ∆ = the set of f ∈ A s.t. A  (f is a function ∧ dom(f ) ∈ Hτ ). For each f ∈ ∆ let d(f ) = that u ∈ H s.t. u = dom(f ) in A. Set: Γ = {hf, xi | f ∈ ∆ ∧ x ∈ π(d(f ))}. It is easily seen by τ -cofinality that each a ∈ A has the form: a = π(f )(x) in A, where hf, xi ∈ Γ. The same holds for hA′ , π ′ i if hA′ , π ′ i is another liftup. But: π(f )(x) ∈ π(g)(y) in A ←→ hx, yi ∈ π({hz, wi | f (z) ∈ g(w) in A}) ←→ π ′ (f )(x) ∈ π ′ (g)(y) in A ! Similarly: π(f )(x) = π(g)(y) in A ←→ π ′ (f )(x) = π ′ (g)(y) in A′ . ∼ Hence there is σ : A ↔ A′ defined by σ(π(f )(x)A ) = π ′ (f )(x)A for hf, xi ∈ Γ. But for any a ∈ A, we have: A  a = ka (0), where ka = {ha, 0i} in A. Thus π(a) = π(ka )(0) in A, where hka , 0i ∈ Γ. Hence σ(π(a)) = π ′ (ka )(0) = π ′ (a). QED(Lemma 3.1) Since the identity is the only isomorphism of a transitive structure onto a transitive structure, we have: Corollary 3.2 Let hA, πi be the liftup hA, πi, where A, A are transitive. Then hA, πi is the unique liftup. ∼ Proof . Let hA′ , π ′ i be a liftup. Let σ : A ↔ A′ s.t. π ′ = σπ. Then A′ is well founded, hence transitive, by solidity. Hence σ = id and π ′ = π, A′ = A.QED(Corollary 3.2) A transitive liftup does not always exist, even when A is transitive. However, a straightforward modification of the ultrapower construction does give us: Lemma 3.3 Let A be a solid model of ZFC− . Let τ > ω, τ ∈ wfc(A) be a cardinal in A and set: H = HτA . Let π : H →Σ0 H cofinally, where H is transitive. Then hA, πi has a liftup hA, πi. A Proof . Define ∆, Γ as above. Let A = h|A|, ∈A , AA 1 , . . . , An i. Define an equality ∗ ∗ ∗ ∗ ∗ model Γ = hΓ, = , ∈ , A1 , . . . , An i by: hf, xi =∗ hg, yi ←→ hx, yi ∈ π({hz, wi | f (z) ∈ g(w) in A}) hf, xi ∈∗ hg, yi ←→ hx, yi ∈ π({hz, wi | f (z) ∈ g(w) in A}) hf, xi ∈ A∗i ←→ x ∈ π({z | f (z) ∈ Ai in A}). A straightforward modification of the usual proof gives us Los’ Theorem for Γ∗ : jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 35 Subcomplete Forcing Γ∗  ϕ[hf1 , x1 i, . . . , hfn , xn i] ←→ (1) ←→ hx1 , . . . , xn i ∈ π({h~zi | A  ϕ[f1 (z1 ), . . . , fn (zn )]}). This is proven by induction on ϕ. The case that ϕ is a primitive formula is immeW diate. We display the induction step for ϕ = ϕ(v1 . . . . , vn ) = v0 ψ(v0 , . . . , vn ). (→) Let Γ∗  ϕ[hf1 , x1 i, . . . , hfn , xn i]. Then Γ∗  ψ[hf0 , x0 i, . . . , hfn , xn i] for some hf0 , x0 i ∈ Γ. Hence hx0 , . . . , xn i ∈ π({h~z i | A  ψ[f0 (z0 ), . . . , fn (zn )}) \ π(d(f0 ) × {h~z i | A  ϕ[f1 (z1 ), . . . , fn (zn )]}) −→ hx1 , . . . , xn i ∈ π({(~z ) | A  ϕ[f1 (z1 ), . . . , fn (zn )]}). (←) Set u = {h~z i | A  ϕ[f1 (z1 ), . . . , fn (zn )]}. Then u ∈ H and h~x i ∈ π(u). In A V W we have ~z y(y, f1 (z1 ), . . . , fn (zn )). Hence, by ZFC− , there is f0 ∈ A s.t. V ~z ψ(f0 (~z ), f1 (z1 ), . . . , fn (zn )) in A. But then hf0 , h~z ii ∈ Γ and hh~x i, x1 , . . . , xn i ∈ π({h~z i | A  ψ[z0 , . . . , zn ]}). Hence Γ∗  ψ[hf0 , h~x ii, hf1 , x1 i, . . . , hfn , xn i]. QED(1) Now let Γ′ = h|Γ′ |, ∈′ , A′1 , . . . , A′n i be the result of factoring Γ∗ by =∗ , the elements being the =∗ -equivalence classes x′ of x ∈ Γ. Since Γ′ satisfies extensionality, ∼ there is an isomorphism σ : Γ′ ↔ A, where A is solid. Set: [f, x] = σ(hf, xi′ ), where − hf, xi ∈ Γ. Then A  ZFC by (1). We now define π : A ≺ A by: Definition For a ∈ A let k = {ha, 0i} in A. Set: π(a) =Df [k, 0]. Then: π : A ≺ A. (2) Proof . A  ϕ[a1 , . . . , an ] ←→ h0 − 0i ∈ {h~z i | A  ϕ[ka1 (z1 ), . . . , kan (zn )]} ←→ h0 − 0i ∈ π({h~z i | A  ϕ[ka1 (z1 ), . . . , kan (zn )]}) ←→ A  ϕ[π(a1 ), . . . , π(an )] by (1). QED(2) Now set: Definition ∆0 = the set of functions f ∈ H. Γ0 = the set of hf, xi s.t. f ∈ ∆0 and x ∈ π(dom(f )). Since π : H → H cofinally, H is the set of π(f )(x) s.t. hf, xi ∈ Γ0 . Now set: Definition H̃ = {[f, x] | hf, xi ∈ Γ0 }. (3) H̃ is “A-transitive” – i.e if a ∈A b ∈ H̃, then a ∈ H̃. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 36 jensen Subcomplete Forcing and L-Forcing Proof . Let a = [f, x], b = [g, y], where hg, yi ∈ Γ0 and hf, xi ∈ Γ. Set: u = {z ∈ d(f ) | f (z) ∈ H}: Then hf, xi ∈∗ hg, yi implies hf, xi =∗ hf ↾ u, xi, where hf ↾ u, xi ∈ Γ0 . QED(3) But for hf, xi, hg, yi ∈ Γ0 we have: [f, x] ∈ [g, y] in A ←→ hx, yi ∈ π({hz, wi | f (z) ∈ g(w)}) ←→ π(f )(x) ∈ π(g)(y). Similarly: [f, x] = [g, y] ↔ π(f )(x) = π(g)(y). Hence there is an isomorphism ∼ σ : hH̃, ∈A i ↔hH, ∈i defined by: σ([f, x]) = π(f )(x) for hf, xi ∈ Γ0 . Hence hH̃, ∈A i is well founded. Since H̃ is A-transitive it follows that H̃ ⊂ wfc(A); hence ∈A ∩H̃ 2 =∈ ∧H 2 by solidity. Hence H̃ is transitive. Thus σ = id and (4) H̃ = H ⊂ wfc(A) and [f, x] = π(f )(x) for hf, xi ∈ Γ0 . But then: (5) [f, x] = π(f )(x) in A for all hf, xi ∈ Γ. Proof . x ∈ π(d(f )), where d(f ) = {x | f (x) = f (x)} = {x | f (x) = (kf (0))(id ↾ d(f ))(x) in A} where kf = {hf, 0i} in A. Hence hx, 0, xi ∈ π({hz, y, wi | f (z) = kf (y)(id ↾ d(f ))(z) in A}. Thus [f, x] = [kf , 0]([(id ↾ d(f )), x]) in A, where: [kf , 0] = π(f ) and [id ↾ d(f ), x] = π(id ↾ d(f ))(x) = x by (4). (6) QED(5) π ↾ H = π, since for a ∈ H we have π(a) = [ka , 0] = π(ka )(0) = kπ(a) (0) = π(a) by (4). Finally, since every a ∈ A has the form π(f )(x) for an x ∈ H, it follows that a ∈ π(rng(f )) in A, where rng(f ) < τ in A. Thus (7) π : A ≺ A τ -cofinally. QED(Lemma 3.3) The above proof yields more than we have stated. For instance: Lemma 3.4 Let π : N →Σ0 N confinally, where N is a ZFC− model and N is transitive. Then π : N ≺ N . (Hence N is a ZFC− model.) Proof . Repeat the above proof with τ = On ∩ N (hence H = N ). All steps go through and we get A = H̃ = N . QED(Lemma 3.4) Lemma 3.5 Let A, A, H, H, τ , π be as in Lemma 3.3. Set τ̃ = On ∩ H. Then τ̃ ∈ wfc(A) and H = Hτ̃A . Proof . By the definition of wfc(A) we have: (∗) If x ∈ A and y ∈ wfc(A) whenever y ∈A x, then x ∈ wfc(A). We consider two cases: July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing jensen 37 Case 1 τ is regular in A. A Claim H = Hπ(τ ) (hence π(τ ) = τ̃ ∈ wfc(A)). Proof . (⊂) is trivial. We prove (⊃). Let x ∈ Hπ(τ ) in A. We claim that x ∈ H. Let x ∈ π(u) in A, where u ∈ A, u < τ in A. Let v = u ∩ Hτ in A. Then v ∈ H = HτA by regularity of τ . But then x ∈ π(v) ∈ H. Hence x ∈ H. QED(Case 1) Case 2 Case 1 fails. Let κ = cf(τ ) in A. Then κ ∈ H. Let f : κ → τ in A be normal and cofinal in τ . Then f ∈ wfc(A) by (∗). Let κ̃ = sup π ′′ κ. Then κ̃ ≤ π(κ) ∈ H. Hence κ̃ ∈ H. Let g = π(f ) ↾ κ̃ in A. It follows easily by (∗) that g ∈ wfc(A). Thus τ̃ = sup g ′′ κ̃ ∈ wfc(A). Claim H = Hτ̃A (⊂) Let x ∈ H. Then x ∈ π(u) where u ∈ H. Hence x ∈ π(u) ∈ Hτ̃ . Hence x ∈ Hτ̃ . A (⊃) Let x ∈ Hτ̃A . Then x ∈ Hπ(ν) for a ν < τ which is regular in A, since τ̃ = sup π ′′ τ and τ̃ is a limit cardinal in A. Let x ∈ π(u) in A, where u ∈ A, u < τ in A. We can choose ν large enough that u < ν in A. Let v = u ∩ Hν in A. Then v ∈ Hν ⊂ H and x ∈ π(v) ∈ H. QED)(Lemma 3.5) An immediate corollary of the proof is: A Corollary 3.6 If τ is regular or cf(τ ) = ω in A. Then τ̃ = π(τ ) and H = Hπ(τ ). Note that if N , N are transitive ZFC− models, τ ∈ N is a cardinal in N and π : N ≺ N τ -cofinally, then π is κ cofinal for every κ ≥ τ which is a cardinal in N . Hence, by Corollary 3.6 we conclude: Corollary 3.7 Let π : N →Σ0 N τ -cofinally, where N , N are transitive, τ ∈ N is a cardinal in N , and N  ZFC− . Let κ ≥ τ be regular in N or cf(κ) = ω in N . S N Then π(κ) = sup π ′′ κ and Hπ(κ) = π(u). N u∈Hκ ⋆⋆⋆⋆⋆ We are now ready to develop the concept of fullness further. We first generalize it as follows: Definition Let N be a transitive ZFC− model. N is almost full iff ω ∈ N and there is a solid A s.t. • A  ZFC− , • N ∈ wfc(A), • N is regular in A – i.e. if f : x ∈ N , x ∈ N , and f ∈ A, then rng(f ) ∈ N . The last condition can be alternatively expressed by: |N | = HτA , where τ = On∩N . Definition A verifies the almost fullness of N iff the above holds. July 21, 2012 15:2 38 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing Clearly every full structure is almost full. By Lemma 3.3 and 3.5 we then have: Lemma 4.1 Let N be almost full. Let π : N →Σ0 N cofinally, where N is transitive. Then N is almost full. (In fact, if A verifies the almost fullness of N and hA, πi is a liftup of hA, πi, then A verifies the almost fullness of N.) Definition Let N be a transitive ZFC− model. δN = the least δ s.t. Lδ (N ) is admissible. By Chapter 1 Corollary 21.1 we then have: Lemma 4.2 If A verifies the almost fullness of N , then LδN (N ) ⊂ wfc(A). Combining this with Lemma 4.1 we get a conclusion that is rich in consequences: Lemma 4.3 Let π : N →Σ0 N cofinally where N is almost full and N is transitive. Let ϕ be a Π1 condition. Let a1 , . . . , an ∈ N . Then LδN (N )  ϕ[N , ~a ] −→ LδN (N )  ϕ[N, π(~a )]. Proof . Let A verify the almost fullness of N and let hA, π̃i be a liftup of hA, πi. We assume: LδN (N )  ψ[N, π(~a )], where ψ is a Σ1 condition, and prove: Claim LδN (N )  ψ[N , ~a ]. Set: ν = the least ordinal s.t. Lν (N )  ψ[N, π(~a )]. Then ν < δN . Noting that A  ψ[N, π(~a )], we see that ν is A-definable, hence has a preimage ν under π̃ AO π̃ /A O ? wfc(A) O ? wfc(A) O ? Lδ (N ) O ν ? N ? Lδ (N ) O ν = π̃(ν) ? /N π Since ν ∈ wfc(A), we conclude that ν ∈ wfc(A). Hence Lν (N )  ψ[N , ~a ]. But Lη (N ) is not admissible for any η ≤ ν. Hence Lη (N ) is not admissible for any QED(Lemma 4.3) η ≤ ν. Hence ν < δN and the conclusion follows. We now combine this with Barwise’ theory. Recall that by a theory or axiomatized language on an admissible structure M we mean a pair hL0 , Ai where L0 is a jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing jensen 39 language (i.e. a set of predicates and constants) in M -finitary predicate logic, and A is a set of axioms in L0 . We defined L = hL0 , Ai to be Σ1 (M ) in parameters p1 , . . . , pn ∈ M iff L0 is ∆1 (M ) in p~ and A is Σ1 (M ) in p~. By Chapter 2 Corollary 4 we get: Lemma 4.4 Let M be admissible. Let L = hL0 , Ai be a theory on M which is Σ1 (M ) in parameters p1 , . . . , pn ∈ M . The statement: ’L is consistent’ is then Π1 (M ) in p~ (uniformly in the Σ1 definition of A from p~ ). Hence Lemma 4.5 Let π : N →Σ0 N cofinally, where N is almost full. Let L be an infinitary theory on LδN (N ) which is Σ1 in parameters N , p1 , . . . , pn ∈ N . Let the p ) by the same definition. If L is consistent, theory L on LδN (N ) be Σ1 in N , π(~ so is L. A typical application is: Corollary 4.6 Let π : N →Σ0 N cofinally, where N is almost full. Let ϕ(v1 , . . . , vn ) be a first order (finite) formula in the N -language with one additional ◦ predicate A. Let card(N ) = τ , card(N ) = τ . Let x1 , . . . , xn ∈ N . If coll(ω, τ ) W W forces AhN , Ai  ϕ[~x ]. Then coll(ω, τ ) forces AhN, Ai  ϕ[π(~x )], (coll(ω, τ ) being the usual conditions for collapsing τ to ω). Proof . Let L be the language on LδN (N ) with the basic axioms. The additional ◦ constant a , and the additional axiom: ◦ hN , a i  ϕ[x1 , . . . , xn ]. Let L have the same definition over LδN (N ) in the parameters π(x1 ), . . . , π(xn ). W By Barwise’ completeness theorem, L is consistent iff coll(ω, τ ) forces AhN , Ai  ϕ[~x ]. Similarly for L, N , π(~x ). The conclusion then follows by Lemma 4.5. QED(Corollary 4.6) The theory of liftups also reveals the import of condition (c) in the definition of “subcomplete”. To this end we prove the interpolation lemma: Lemma 5.1 Let π : N ≺ N where N is a transitive ZFC− model and N is S transitive. Let τ be a cardinal in N . Set: H = HτN and H̃ = {π(u) | u ∈ N and u < τ in N }. Then: (a) The transitive liftup hÑ , π̃i of hN , π ↾ Hi exists. (b) There is σ : Ñ ≺ N s.t. σπ̃ = π and σ ↾ H̃ = id. (c) σ is the unique σ ′ : Ñ →Σ0 N s.t. σ ′ π̃ = π and σ ′ ↾ π̃ = id, where τ̃ = On ∩ H̃. Proof . Let hA, π̃i be a liftup of hN , π ↾ Hi. Letting Γ be as in the proof of Lemma 3.3 we see that each y ∈ A has the form π̃(f )(x) in A for some hf, xi ∈ Γ. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 40 Subcomplete Forcing and L-Forcing Moreover: A  ϕ[π̃(f1 )(x1 ), . . . , π̃(fn )(xn )] ←→ ←→ hx1 , . . . , xn i ∈ π({h~z i | N  ϕ[f1 (z1 ), . . . , fn (zn )]}) ←→ N  ϕ[π(f1 )(x1 ), . . . , π(fn )(xn )]. Hence there is σ : A ≺ N defined by: σ(π̃(f )(x)) = π(f )(x) for hf, xi ∈ Γ. Thus A is well founded, hence transitive by solidity. This proves (a), (b). We now prove (c). Let σ ′ be as in (c). Since π : N ≺ Ñ τ -cofinally, it follows that any y ∈ Ñ has the form π(f )(ν) for an hf, νi ∈ Γ s.t. dom(f ) ⊂ τ . Hence σ ′ (y) = π(f )(ν) = σ(y). QED(Lemma 5.1) Just as in the proof of Lemma 3.4 we can repeat this using τ = On ∩ N , getting: Lemma 5.2 Let π : N ≺ N where N , N are transitive ZFC− models. Set: S π(u). (Hence π : N ≺ Ñ cofinally.) Then Ñ ≺ N . Ñ = u∈N We now utilize this to examine the meaning of (c) in the definition of “subcomplete”. − Lemma 5.3 Let σ : N ≺ N where N = LA α is a ZFC model and N is transitive. Let σ(δ) = δ, where δ is a cardinal in N . Set C = CδN (rng(σ)), H = (Hδ+ )N , S H̃ = σ(u). Let hÑ , σ̃i be the liftup of hN , σ ↾ Hi and let k = Ñ ≺ N s.t. u∈H kσ̃ = σ and k ↾ H̃ = id. Then C = rng(k). Proof . (⊂) rng(σ) ⊂ rng(k) and σ ⊂ rng(k). (⊃) Let x ∈ rng(k), x = k(x̃) where x̃ ∈ σ̃(u), u ∈ N , u < δ + in N . Let f ∈ N , onto f : δ −→ u. Then x = kσ̃(f )(ν) = σ(f )(ν) for a ν < δ. Hence x ∈ C. QED(Lemma 5.3) ∼ Stating this differently, we can recover Ñ , k from C by the definition: k : Ñ ↔ C, where Ñ is transitive. We can then recover σ̃ from C by σ̃ = k −1 · σ. If we now have another σ ′ : N ≺ N s.t. σ ′ (δ) = δ and C = CδN (rng(σ ′ )), then hÑ , σ̃ ′ i is the liftup of hN , σ ′ ↾ Hi, where σ̃ ′ = k −1 σ ′ . Thus σ = kσ̃, σ ′ = kσ ′ where σ̃, σ̃ ′ are determined entirely by σ ↾ H, σ ′ ↾ H, respectively. Hence Corollary 5.4 Let σ, σ ′ be as above. Let τ ∈ N be regular in N s.t. τ > δ and σ(τ ) = σ ′ (τ ). Then sup σ ′′ τ = sup σ ′ ′′ τ . Proof . Let k(τ̃ ) = σ(τ ) = σ ′ (τ ). Then τ̃ = sup σ̃ ′′ τ = sup σ̃ ′ ′′ τ , since σ̃, σ̃ ′ are τ -cofinal and τ is regular in N . But then: sup σ ′′ τ = sup σ ′ ′′ τ = sup k ′′ τ̃ . QED(Corollary 5.4) A similar argument yields: Corollary 5.5 Let τ = On ∩ N , where σ, σ ′ are as above. Then sup σ ′′ τ = sup σ ′ ′′ τ = sup k ′′ τ̃ , where τ̃ = On ∩ Ñ . Our original version of (c) was weaker, and can be stated as: jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 41 Subcomplete Forcing (c′ ) Let s = hs0 , λ1 , . . . , λn i and s = hs0 , λ1 , . . . , λn i where λi > δ is regular in N . Let λ0 = On ∩ N . Then sup σ ′′ λi = sup σ ′ ′′ λi for i = 0, . . . , n. This is, of course, an immediate consequence of the above two corollaries. The weaker definition of ’subcomplete’ should not be forgotten, since we might someday encounter a forcing which sastisfies the weaker version but not the stronger one. That has not happened to date, however, and in fact our original verifications of (c′ ) turned essentially on first verifying (c). Before leaving the topic of τ -cofinal embeddings, we mention that these concepts can be applied to structures that are not ZFC− models. For our purposes it will suffice to deal with the class of smooth models: Definition Let N be a transitive model. N is smooth iff either N  ZFC− or else S there is a sequence hhNi , αi i | i < λi of limit length s.t. N = Ni and Nj  ZFC− , i N Nj is transitive, and Ni ∈ Nj s.t. αi is regular in Nj and Ni = Hαij for i < j < λ. Then: Lemma 5.6 If N is smooth, N transitive, and π : N →Σ0 N cofinally, then N is smooth. Proof . If N  ZFC− , this is immediate from the foregoing. Otherwise there is a sequence hhN i , αi i which verifies the smoothness of N . Set Ni = π(N i ), αi = π(αi ). Then hhNi , αi i | i < λi verifies the smoothness of N . QED(Lemma 5.6) Note It does not follow that π : N ≺ N . The concepts “τ -cofinal” and “liftup” are defined as before, and it follows as before that if N is smooth, τ is a cardinal in N and π : HτN →Σ0 H cofinally, then hN , πi has at most one transitive liftup. Lemma 5.7 Let π : N →Σ0 N τ -cofinally, where N is smooth. Let κ ∈ N be N regular in N , where κ > τ . Let H = HκN , H = Hπ(κ) . Then π ↾ H : H →Σ0 H τ -cofinally. Proof . Exactly as in Case 1 of Lemma 3.5. Lemma 5.8 Let hhN i , αi i | i < λi verify the smoothness of N . Let τ ∈ N be a cardinal. Let π : HτN →Σ0 H cofinally. The transitive liftup of hN , πi exists iff for each i s.t. τ < αi the transitive liftup of hN i , πi exists. Proof . (→) Let hN, πi be the liftup of hN , πi. Set: αi = π(αi ), Ni = π(N i ). Then hNi , π ↾ N i i is the liftup of hN i , πi. (←) Let hNi , πi i be the liftup of hN i , πi for τ < αi . By Lemma 5.7 we have: S πj ↾ N i : N i → Ni τ -cofinally. Hence πj ↾ N i = πi and we can set: π = πi . i π : N →Σ0 N is then τ -cofinal and π = π ↾ H. QED(Lemma 5.8) Lemma 5.9 Let N be smooth and π : N →Σ0 N , where N is transitive. Let τ be a cardinal in N . Set H = HτN . Then: July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 42 jensen Subcomplete Forcing and L-Forcing (a) The transitive liftup hÑ , π̃i of hN , π ↾ Hi exists. S (b) There is σ : Ñ →Σ0 N s.t. σπ̃ = π and σ ↾ H̃ = id, where H̃ = π(u). u∈H (c) σ is the unique σ : Ñ →Σ0 N s.t. σπ̃ = π and σ ↾ τ̃ = id, where τ̃ = On ∩ H̃. Proof . Case 1 N  ZFC− . S Set: N ′ = π(u). Then π : N →Σ0 N ′ cofinally. Hence π : N ≺ N ′ ⊂ N and we u∈N apply our previous lemmas. Case 2 Case 1 fails. Let hhN i , αi i | i < λi verify the smothness of N . Assume w.l.o.g. that τ ∈ N 0 . (Hence H = HτN i for all i < λ.) (a) follows by Lemma 5.8. Moreover hN i , Ñi i is the liftup of hN , π ↾ Hi by Lemma 5.7, where Ñi = π(N i ). Let σi : Ñi →Σ0 π(N i ) S be defined by σi π̃i = π ↾ N i , σi ↾ H̃ = id. Set σ = σi . Then σ : Ñ →Σ0 N and i σπ̃ = π, σ ↾ H̃ = id. This proves (b). But σi is unique s.t. σi : Ñi →Σ0 π(N i ), σi π̃ = π ↾ N i and σi ↾ τ̃ = id. Hence σ ↾ Ñi = σi for i < λ if σ is as in (c). This proves (c). QED(Lemma 5.9) 3.3 Examples We are now ready to prove that some specific forcings are subcomplete. Since these forcings will be presented as sets of conditions rather than Boolean algebras, we set: Definition Let P be a set of conditions. VP =Df VBA(P) , δ(P) =Df δ(BA(P)) where BA(P) is the canonical Boolean algebra over P as defined in Chapter 0. We may refer to the elements of VP as ’P-names’. We note: − Fact 1 Let N = LA τ be a ZFC model. Let BA(P) ∈ Hθ ⊂ N . Let δ ⊂ C ≺ N , where BA(P) ∈ C and δ = δ(P). Then for each p ∈ P there is q ∈ C ∩P s.t. [q] ⊂ [p]. (Hence every set predense in C ∩ P is predense in P.) Proof . Let B = BA(P). By definition there are f, ∆ ∈ Hθ s.t. ∆ is dense in B and f : δ ↔ ∆. Hence there are such f, ∆ ∈ C. But ∆ ⊂ C, since δ ⊂ C. Let p ∈ P. There is b ∈ ∆ s.t. b ⊂ [p]. Hence there is q ∈ C ∩ P s.t. [q] ⊂ b, since C ≺ N . Hence [q] ⊂ [p]. QED(Fact 1) Our first example is Prikry forcing. Lemma 6.1 Prikry forcing is subcomplete. Proof . Let U be a normal ultrafilter on a measurable cardinal κ. We define the Prikry forcing determined by U to be the set P = PU consisting of all pairs hs, Xi July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 43 Subcomplete Forcing s.t. X ∈ U and s ⊂ κ is finite. The extension relation ≤P is defined by: hs, Xi ≤ ht, Y i iff X ⊂ Y, s ⊃ t, and t = lub(t) ∩ s. P does not collapse cardinals or add new bounded subsets of κ If G is P-generic, the P-sequence added by G is [ W S = SG = {s | Xhs, Xi ∈ G}. Then S is unbounded in κ and has order type ω. G is, in turn, definable from S by: G = GS = {hs, Xi ∈ P | s = S ∩ lub(s) ∧ S \ s ⊂ X}. Definition We call S ⊂ κ a P-sequence (or Prikry sequence) iff S = SG for some P-generic G. The following characterization of Prikry sequences is well known: Fact 2 S is a Prikry sequence iff S ⊂ κ has order type ω and is almost contained W in every X ∈ U (i.e. ν < κ S \ ν ⊂ X). κ We now prove that P is subcomplete. To this end we let θ > 22 and let N = LA τ be a ZFC− model s.t. τ > θ and Hθ ⊂ N . Furthermore we assume that σ : N ≺ N where N is countable and full. We also suppose that σ(θ, U , P, s) = θ, U, P, s. Hence σ(B) = B, where B = BA(P) and B = BA(P) in N . We must show: Main Claim σ ′ ∈ V[G] s.t. (a) (b) (c) (d) There is p ∈ P s.t. whenever G ∋ p is P-generic. Then there is σ′ : N ≺ N , σ ′ (θ, U , P, s) = θ, U, P, s, CδN (rng σ ′ ) = CδN (rng σ), where δ = δ(P). σ ′ ′′ G ⊂ G. Note that if we set: S = SG and S = SG in N [G], then (d) becomes equivalent to: (d′ ) σ ′ ′′ S = S. Let C = CδN (rng σ). Using Fact 1 we get: (1) Let X ∈ U . Then there is Y ∈ C ∩ U s.t. Y ⊂ X. Proof . Suppose not. Then for each ν < κ the set ∆ν is dense in P ∩ C where ∆ν = {hs, Y i ∈ P ∩ C | s \ ν 6⊂ X}. Hence ∆ν is predense in P by Fact 1. Let G be P-generic. Then G ∩ ∆ν 6= ∅ for ν < κ. Hence SG is not almost contained in X. Contradiction! by Fact 2. QED(1) Hence: July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 44 Subcomplete Forcing and L-Forcing (2) S is a P-generic sequence iff S has order type ω and is almost contained in every X ∈ C ∩ U. (3) δ ≥ κ, T since otherwise C < κ and C ∩ U would have a minimal element Y = (C ∩ U ). ∼ Definition We define N0 , k0 , σ0 by: k0 : N0 ↔ C, where N0 is transitive σ0 = k0−1 ◦ σ. We also set: Θ0 , P0 , U0 , s0 = σ0 (θ, P, U , s). By Chapter 3.2, however, we have an alternative characterization: (4) Let σ0 (δ) = δ, ν = δ +N , H = HνN . Then hN0 , σ0 i is the liftup of hN , σ ↾ Hi. Moreover k0 is defined by the condition: k0 : N0 ≺ N, k0 σ0 = σ, k0 ↾ ν0 = id, where ν0 = sup σ ′′ ν. Since ν is regular in N , we conclude: (5) σ0 is a ν-cofinal map and σ0 (ν) = ν0 . Definition α0 = δN0 = the least α s.t. Lα (N0 ) is admissible. Our Main Claim will reduce to the assertion that a certain language L0 on Lα0 (N0 ) is consistent. We define: Definition L0 is the language on Lα0 (N0 ) with: Predicate: ∈ ◦ ◦ Constants: S , σ , x (x ∈ Lα0 (N0 )) Axioms: • Basic axioms and ZFC− ◦ • S is P0 -sequence over N0 ◦ • σ : N ≺ N 0 κ-cofinally, where σ(κ) = κ ◦ • σ (θ, P, U , s) = θ 0 , P0 , U 0 , s0 • ◦ ◦ σ ′′ S = S . We first show that L0 is consistent. To this end we define: Definition hN1 , σ1 i = the liftup of hN , σ ↾ HκN i. k1 = the unique k : N1 ≺ N0 s.t. kσ1 = σ0 and k ↾ κ1 = id, where κ1 = sup σ ′′ κ. θ1 , P1 , U1 , s1 = σ1 (θ, P, U , s), S1 = σ1 ′′ S. Note that κ1 = σ1 (κ), since σ1 is κ-cofinal into N1 and κ is regular in N . Then: (6) (a) (b) (c) (d) S1 is a P1 -sequence over N1 , σ1 : N ≺ N1 κ-cofinally, σ1 (θ, P, U , s) = θ1 , P1 , U1 , s1 , σ1 ′′ S = S1 . Proof . (b)–(d) are immediate. (a) follows by: jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 45 Subcomplete Forcing Claim Let X ∈ U1 . Then S1 is almost contained in X. T Proof . Let X ∈ σ1 (w), where w < κ in N . Then Y = (U ∩ w) is almost contained in every z ∈ U ∩ w and Y ∈ U . Hence Y = σ1 (Y ) is almost contained in every Y ∈ U1 ∩ σ1 (w). In particular, Y is almost contained in X. But S1 is almost contained in Y . QED(6) Now let: α1 = δN1 = the least α s.t. Lα (N1 ) is admissible. Let L1 be the language Lα1 (N1 ) which is defined as L0 was defined on Lα0 (N0 ), substituting θ1 , P1 , U1 , s1 , κ1 for θ0 , P0 , U0 , s0 , κ0 . Then (7) L1 is consistent. Proof . hHκ , S1 , σ1 i models L1 by (6). QED(7) Note, however, that: NO 1 σ1 k1 / > N0 || | | || σ || 0 N where all maps are cofinal and all models are almost full. Then L0 is Σ1 (Lα0 [N0 ]) in N0 and the parameters: κ, P0 , κ, N , θ, P, U , s, θ0 , P0 , U0 , s0 . But L1 is Σ1 (Lα1 [N1 ]) in N1 and the k1 -preimages of these parameters by the same Σ1 -formula. Since N1 is almost full and k1 : N1 ≺ N0 cofinally, we conclude by Lemma 4.5: (8) L0 is consistent. From this we now derive the Main Claim: Work in a generic extension V[F ] of V in which Lα0 [N ] is countable. Then L0 has a solid model ◦ ◦ A = h|A|, S A , σ A i. ◦ ◦ Set S = S A , σ ′ = k0 ◦ σ A . Then (8) (a) (b) (c) (d) (e) σ′ : N ≺ N , S is P-generic over V, σ ′ (θ, P, U , s) = θ, P, U, s, CδN (rng σ ′ ) = C, S = σ ′ ′′ S. ◦ ◦ Proof . (a), (c) are immediate. To see (d) note that N0 = CδN0 (rng σ A ), since σ A is ◦ κ-cofinal and δ ≥ κ = σ A (κ). Hence: ◦ C = k0 ′′ N0 = CδN (rng k0 ◦ σ A ). July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 46 Subcomplete Forcing and L-Forcing Since k0 ↾ (κ + 1) = id we have U ∩ N0 = U ∩ C. Hence (b) follows by (2). (e) ◦ follows by σ ′ ↾ κ = σ A ↾ κ. QED(9) We have almost proven the Main Claim, the only problem being that σ ′ is not necessarily an element of V[S]. We now show: (10) There is σ ′ ∈ V[S] satisfying (9). Proof . Work in V[S]. Let µ be regular in V[S] s.t. N ∈ Hµ . Set: M = hHµ , N, S, θ, P, U, s, σi. We define a language L2 on the admissible structure M as follows: Definition L2 is the language on M with Predicate: ∈ ◦ Constants: σ , x, (x ∈ M ) Axioms: • ZFC− and basic axioms ◦ • σ :N ≺N ◦ • σ (θ, P, U , s) = θ, P, U , s ◦ N • Cδ (rng σ ) = C • ◦ S = σ ′′ S L2 is clearly consistent, since hM, σ ′ i is a model of L2 in V[F ], where σ ′ is defined as above. Now let π : M̃ ≺ M , where M̃ is countable and transitive. Let L̃2 be the language on M̃ with the same Σ1 definition, replacing all parameters by their preimages V[S] under π. Then L̃2 is consistent. Since M̃ ∈ Hω1 = HωV1 , it follows that L̃2 has a ◦ solid model à in V. Let σ̃ = σ à and set: σ ′ = π ◦ σ̃. The verification of (9) is then straightforward. QED(10) But, since S is a Prikry sequence, there must be p ∈ GS which forces the existence of such a σ ′ . This proves the Main Claim. QED(Lemma 6.1) Lemma 6.2 Assume CH. Then Namba forcing is subcomplete. Proof . We first define Namba forcing. The set ω2<ω of monotone finite sequences in ω2 is a tree ordered by inclusion. The set N of Namba conditions is the collection of all subtrees T 6= ∅ of ω2<ω s.t. T is downward closed in ω2<ω and for each s ∈ T the set {t | s ≤T t} has cardinality ω2 . The extension relation ≤N is defined by: T ≤ T ′ ←→Df T ⊂ T ′ . ST If G is N-generic, then S = G is a cofinal map of ω into ω2V . We rite S = SG and call any such S a Namba sequence. G is then recoverable from S by: V G = GS = {T ∈ N | n < ω S ↾ n ∈ T }. It is known that, if CH holds, then Namba forcing adds no reals. We shall also make use of the following fact, which is proven in the Appendix to [DSF]: jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 47 Subcomplete Forcing Fact Let S be a Namba sequence. Let S ′ ∈ V[S] be a cofinal ω-sequence in ω2V . Then S ′ is a Namba sequence and V[S ′ ] = V[S]. Note that δ(N) ≥ ω2 , since otherwise ω2 would not be collapsed. ω2 − We now turn to the proof. Let θ > 22 . Let N = LA τ be a ZFC model s.t. τ > θ and Hθ ⊂ N . Let σ : N ≺ N where N is countable and full. Let σ(θ, N, s) = θ, N, s. Let G be N-generic over N . It suffices to show: Main Claim There is p ∈ N s.t. whenever G ∋ p is N-generic, then there is σ ′ ∈ V[G] with: (a) (b) (c) (d) σ′ : N ≺ N , σ ′ (θ, N, s) = θ, N, s, CδN (rng σ ′ ) = CδN (rng σ) where δ = δ(N), σ ′ ′′ G ⊂ G. Note We shall actually prove a stronger form of (c): CωN2 (rng σ ′ ) = CωN2 (rng σ). Note (d) can equivalently be replaced by: σ ′ ′′ S = S, where S = SG , S = SG . Definition Set C = CωN2 (rng σ). Define k0 by: ∼ k0 : N0 ↔ C, where N0 is transitive, σ0 = k0−1 ◦ σ, θ0 , N0 , s0 = σ0 (θN, s). Just as before we get: (1) hN0 , σ0 i is the liftup of hN , σ ↾ HωN3 i, k0 is the unique k : N0 ≺ N s.t. k0 σ0 = σ and k0 ↾ ω3N0 = id, (where ω3N0 = sup σ0 ′′ ω3N ). Now let α0 be the least α s.t. Lα (N0 ) is admissible. We define a language L0 on Lα0 (N0 ) as follows: Definition L0 is the language on Lα0 (N0 ) with: Predicate: ∈ ◦ Constants: σ , x (x ∈ Lα0 (N0 )) Axioms: • Basic axioms and ZFC− ◦ N • σ : N ≺ N 0 ω 2 -cofinally ◦ • σ (θ, N , s) = θ0 , N0 , s0 . (2) L0 is consistent. Proof . Let hN1 , σ1 i be the liftup of hN , σ ↾ HωN2 i. Define k1 : N1 ≺ N0 by: k1 σ1 = σ0 , k1 ↾ γ1 = id, where γ1 = sup σ ′ ω2N = σ1 (ω2N ). Let L1 be the corresponding language on Lα1 (N1 ), where α1 = δN1 . Just as before it suffices to show that L1 is consistent. This clear, however, since hHω2 , σ1 i is a model. QED(2) July 21, 2012 15:2 48 World Scientific Book - 9.75in x 6.5in jensen Subcomplete Forcing and L-Forcing Now let S ′ be a Namba sequence. Work in V[S ′ ]. Let µ be a regular cardinal in V[S ′ ] with N ∈ Hµ . Set: M = hHµ , N, σ, N, si. Let π : M̃ ≺ M , where M̃ is transitive and countable. Then M̃ ∈ Hω1 ⊂ V in V [S ′ ]. Let π(Ñ, σ̃, Ñ , L̃, k̃, C̃) = N, σ, N, L0 , k0 , C0 . ◦ Let A ∈ V be a solid model of L̃. Set σ̃ = k̃ ◦ σ A ; σ ′ = π ◦ σ̃. It follows easily that: (3) (a) σ ′ : N ≺ N (b) σ ′ (θ, N, s) = θ, N, s (c) CωN1 (rng σ ′ ) = C Now let S = SG and set: S = σ ′ ′′ S. Then S ∈ V[S ′ ] is a cofinal ω-sequence in ω2V ; hence: (4) S is a Namba sequence and V[S] = V[S ′ ]. (Hence σ ′ ∈ V[S].) But we know: (5) S = σ ′ ′′ S. Let G = GS . There is then a p ∈ G which forces the existence of a σ ′ ∈ V[S] satisfying (3), (5). This proves the Main Claim. QED(Lemma 6.2) Now let κ > ω1 be a regular cardinal. Let A ⊂ κ be a stationary set of ω-cofinal ordinals. Our final example is the forcing PA which is designed to shoot a cofinal normal sequence of order type ω1 through A: Definition PA is the set of normal functions p : ν + 1 → A, where ν < ω1 . The extension relation is defined by: p≤q in PA ←→Df q ⊂ p. S Clearly, if G is PA -generic, then G : ω1 → A is normal and cofinal in κ. PA adds no new countable subsets of the ground model. If, however, {λ < κ | cf(λ) = ω∧λ∈ / A} is stationary, then PA will not be a complete forcing. Lemma 6.3 PA is subcomplete. κ Proof . Clearly δ(PA ) ≥ κ, since otherwise κ would remain regular. Now let θ > 22 . − Let N = LB τ be a ZFC model s.t. τ > θ and Hθ ⊂ N . Let σ : N ≺ N where N is countable and full. Let σ(θ, P, A, κ, s) = θ, PA , A, κ, s. Let G be P-generic over N . It suffices to show: Main Claim There is p ∈ P s.t. whenever G ∋ p is PA -generic, there is σ ′ ∈ V[G] s.t. (a) σ ′ : N ≺ N , July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing jensen 49 (b) σ ′ (θ, P, κ, A, s) = θ, P, κ, A, s, (c) CκN (rng σ ′ ) = CκN (rng σ), (d) σ ′ ′′ G ⊂ G. (1) Let σ ′ ∈ V satisfy (a), (b), (c) and (e) sup σ ′ ′′ κ ∈ A. Then the Main Claim holds. S G. Then F is a cofinal normal map of ω1N into A, where Proof . Let F = σ(A) = A. Define p ∈ PA by: ( σ ′ F (ξ) for ξ < ω1N , p(ξ) = sup σ ′ ′′ κ for ξ = ω1N . Clearly p ≤ σ ′ (q) for q ∈ G. Hence if G ∋ p is generic, then σ ′ ′′ G ⊂ G. QED(1) We must produce a σ ′ satisfying (a), (b), (c) and (e). For ξ < κ set: Cξ = CξN (rng σ). Set: D = {τ < κ | τ = κ ∩ Cτ }. Then D is club in κ. Hence then is κ0 ∈ D ∩ A. Set: ∼ Definition k0 : N0 ↔ Cκ0 , where N0 is transitive; σ0 = k0−1 ◦ σ; θ0 , P0 , s0 , A0 = σ0 (θ, P, s, A). (Hence κ0 = σ0 (κ).) We again let α0 = δN0 be least s.t. Lα0 (N0 ) is admissible and define: Definition L0 is the language on Lα0 (N0 ) with: Predicate: ∈ ◦ Constants: σ , x (x ∈ Lα0 (N0 )) Axioms: • Basic axioms and ZFC− ◦ • σ : N ≺ N 0 κ-cofinally ◦ • σ (θ, P, s, κ, A) = θ 0 , P0 , s0 , κ0 , A0 . (2) L0 is consistent. Proof . Let hN1 , σ1 i be the liftup of hN , σ ↾ HκN i. Note that σ ↾ HκN = σ0 ↾ HκN . Hence there is k1 : N1 ≺ N0 defined by: k1 σ1 = σ0 , k1 ↾ κ1 = id, where κ1 = σ1 (κ) = sup σ ′′ κ. Let L1 be the corresponding language on Lα1 (N1 ), where α1 = δN1 . By the usual argument it suffices to show that L1 is consistent: Since N0 = CκN00 (rng σ0 ), we can conclude that σ0 : N ≺ N0 is κ+N -cofinal. Hence: NO 1 σ1 N k0 / N0 |> | || || σ || 0 July 21, 2012 15:2 50 World Scientific Book - 9.75in x 6.5in jensen Subcomplete Forcing and L-Forcing where all maps are cofinal and all structures are almost full. L1 is trivially consistent, however, since hHκ , σ1 i models L1 . QED(2) Now let M = hHκ , N0 , κ0 , A0 , σ0 i. Let π : M̃ ≺ M s.t. M̃ is countable and transitive. Let π(L̃) = L0 . Then L̃ is a consistent language on Lα̃ (Ñ ) = π −1 (Lα0 (N0 )). Hence L̃ has a solid model A. Set: ◦ σ ′ = k0 ◦ π ◦ σ A . Then σ ′ satisfies (a), (b), (c), (e) of (1). QED(Lemma 6.3) July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen Chapter 4 Iterating subcomplete forcing The two step iteration theorem for subcomplete forcing says that if A is subcomplete and ◦ A is subcomplete, B ◦ then A ∗ B is subcomplete. Equivalently: Theorem 1 Let A ⊆ B where A is subcomplete and ◦ A B̌/G ◦ ◦ is subcomplete. Then B is subcomplete. (Note The definitions of A∗ B, B̌/G and other Boolean conventions employed here ◦ can be found in Chapter 0. G is the canonical generic name – i.e. ◦ ◦ A G is Ǎ-generic over V̌, and [[ǎ ∈ G]] = a for a ∈ A.) Proof of Theorem 1. Let θ be big enough that θ verifies the subcompleteness of A and: ◦ A θ̌ verifies the subcompleteness of B̌/G. − Let N = LA τ be a ZFC model s.t. Hθ ⊂ N and τ > θ. Let σ : N ≺ N where N is countable and sound. Let: σ(θ, A, B, s) = θ, A, B, s where s ∈ N . Let G be B-generic over N . We must find b ∈ B \ {0} s.t. whenever G ∋ b is B-generic, there is σ ′ ∈ V[G] satisfying (a)–(d) in the definition of subcompleteness. Let G0 = G ∩ A. Then G0 is A-generic over N . Since θ verifies the subcompleteness ◦ of A, there exist a ∈ A \ {0}, σ 0 ∈ VA s.t. whenever G0 ∋ a is A-generic and ◦ 0 ′ σ0 = σ G 0 , then (a)–(d) hold with A, G0 , A, G0 , σ0 in place of B, G, B, G, σ . Let ∗ ∗ B = B/G0 . Let G0 ∋ a be A-generic. Set: B = B/G0 . Clearly, σ0 extends to σ0∗ s.t σ0∗ : N [G0 ] ≺ N [G0 ] and 51 σ0∗ (G0 ) = G0 July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 52 Subcomplete Forcing and L-Forcing A,G0 ∗ A,G0 ∗ In other words, σ0∗ : N ∗ ≺ N ∗ where: N = LA , N = LA . τ , N = Lτ τ , N = Lτ V[G0 ] ∗ ∗ ∗ V ∗ Note that Hθ = Hθ [G0 ] ⊂ N . Moreover G is B -generic over N where: G∗ = G/G0 = {b/G0 | b ∈ G}. Clearly σ0∗ (θ, A, B, B∗ , s) = θ, A, B, B∗ , s. Since θ verifies the subcompleteness of B∗ in V[G0 ], we conclude that there is b∗ ∈ B∗ s.t. whenever G∗ ∋ b∗ is B∗ -generic over V[G0 ], then there is σ ∗ ∈ V[G0 ][G∗ ] with: (a∗ ) (b∗ ) (c∗ ) (d∗ ) σ∗ : N ∗ ≺ N ∗ , σ ∗ (θ, A, B, B∗ , s) = θ, A, B, B∗ , s ∗ ∗ CδN∗ (rng(σ ∗ )) = CδN∗ (rng(σ0∗ )), where δ ∗ = δ(B∗ ). σ ∗ ′′ G∗ ⊂ G∗ . Note that G = G0 ∗G∗ =Df {b ∈ B | b/G0 ∈ G∗ }. Set G = G0 ∗G∗ . Set σ ′ = σ ∗ ↾ N . Then σ ′ ∈ V[G] = V[G0 ][G∗ ]. We show: Claim σ ′ satisfies: (a) (b) (c) (d) σ′ : N ≺ N σ ′ (θ, A, B, s) = θ, A, B, s CδN (rng(σ ′ )) = CδN (rng(σ)), where δ = δ(B). σ ′ ′′ G ⊂ G. We note first that the claim proves the theorem, since G is B-generic and there must, therefore, be a b ∈ G which forces the existence of such a σ ′ . We now prove the claim. (a), (b), (d) are immediate. We prove (c). Note that δ ≥ δ ∗ , we have: ∗ ∗ CδN (rng(σ ∗ )) = CδN (rng(σ0∗ )). Since CδN (rng(σ0 )) = CδN (rng(σ)), it suffices to show: ∗ (1) CδN (rng(σ ′ )) = N ∩ CδN (rng(σ ∗ )) ∗ (2) CδN (rng(σ0 )) = N ∩ CδN (rng(σ0∗ )). We proof (1), the proof of (2) being virtually identical. (⊂) is trivial. We prove (⊃). ∗ Let x ∈ N ∩ CδN (rng(σ ∗ )). Then x is N [G0 ]-definable in ξ, σ ∗ (z), G0 , where ξ < δ, z ∈ N . But, letting t ∈ N A s.t. hz, G0 i = tG0 , we have: hσ ∗ (z), G0 i = σ ∗ (hz, G0 i) = σ ∗ (tG0 ) = σ ′ (t)G0 . Hence: x = that x s.t. N [G0 ]  ϕ[x, ξ, σ ′ (t)G0 ]. But since σ ′ (B) = B, we have: σ ′ (δ) = δ, where δ = δ(B). Since δ ≥ δ(A), there is f ∈ N mapping δ onto a dense subset of A. Hence σ ′ (f ) maps δ onto jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 53 Iterating subcomplete forcing a dense subset of A. Hence there is ν < δ s.t. σ ′ (f )(ν) ∈ G0 and σ ′ (f )(ν) ˇ σ ′ (t)). Thus: x = that x s.t. σ ′ (f )(ν) N ϕ(x̌, ξ, ˇ σ ′ (t)). Hence forces ϕ(x̌, ξ, A N ′ x ∈ Cδ (rng(σ )). QED(Theorem 1) The proof of Theorem 1 shows more than we have stated. We can omit the assumption that A is subcomplete and omit the map σ, assuming, however, that ◦ ◦ B̌/G0 is subcomplete, where G0 is the canonical A-generic name. Let θ be big enough that A A ◦ θ̌ verifies the subcompleteness of B̌/G0 . Let N be as before and let N be countable and full. Suppose that a ∈ A \ {0} and ◦ ◦ 0 σ ∈ VA are given s.t. whenever G0 ∋ a is A-generic and σ0 = σ G 0 , then • σ0 : N ≺ N • σ0 (θ, A, B, s) = θ, A, B, s • σ0 ′′ G0 ⊂ G0 . Our proof then yields a b∗ ∈ B∗ = (B/G0 ) \ {0} s.t. if G ⊃ G0 is B-generic and b∗ ∈ G∗ = G/G0 = {c/G0 | c ∈ G}, then there is σ ′ ∈ V[G] s.t. (a), (b), (d) and: (c′ ) CδN (rng(σ ′ )) = CδN (rng(σ0 )) hold, where δ = δ(B). We can improve on this still further. Suppose that t ∈ VA ˇ . This means that tG0 ∈ N whenever G ∋ a is A-generic. We s.t. and a t ∈ N 0 can then select our b∗ so as to force: (e*) σ ∗ (tG0 ) = σ0 (tG0 ). in addition to (a*)–(d*). It then follows that: (e′ ) σ ′ (tG0 ) = σ0 (tG0 ). Since, whenever G0 ∋ a is A-generic, we can find a b∗ ∈ B/G0 forcing (a), (b), (d), ◦ (c′ ), (e′ ), we conclude that there is b0 ∈ VA s.t. a forces b∗ = b G0 to have these properties. We may assume w.l.o.g. that ◦ A ◦ ◦ b ∈ B̌/G0 and [[ b 6= 0]]A = a. By ◦ ◦ Chapter 0, Fact 4 there is then a unique b ∈ B s.t. A b̌/G0 = b . Letting h = hA,B T be defined as in Chapter 0 by h(c) = {a ∈ A | c ⊂ a} for a ∈ B, we conclude by Chapter 0, Fact 3 that: ◦ ◦ h(b) = [[b̌/G0 6= 0]] = [[ b 6= 0]] = a. Clearly, if G ∋ b is B-generic, then G0 ∋ a is A-generic, where G0 = G ∩ A. Thus ◦ b/G0 = b G0 = b∗ has the above properties and (a), (b), (d), (c′ ),(e′ ) hold. Putting all of this together, we get a very useful technical lemma: Lemma 1.1 Let A ⊆ B and let: B ∈ Hθ and: ◦ A B̌/G is subscomplete. Let θ be big enough that ◦ A − θ̌ verifies the subcompleteness of B̌/G. Let N = LA τ be a ZFC July 21, 2012 15:2 54 World Scientific Book - 9.75in x 6.5in jensen Subcomplete Forcing and L-Forcing model s.t. Hθ ⊂ N and θ < τ . Let N be countable and full. Let A ⊆ B in N , where ◦ G is B-generic over N . Set: G0 = G ∩ A. Suppose that a ∈ A \ {0}, σ 0 ∈ VA s.t. ◦ 0 whenever G0 ∋ a is A-generic and σ0 = σ G 0 , then: (i) (ii) (iii) (iv) σ0 : N ≺ N σ0 (θ, A, B, s) = θ, A, B, s σ0 ′′ G0 ⊂ G0 tG 0 ∈ N . ◦ Let h = hA,B . Then there are b ∈ B \ {0}, σ ∈ VB s.t. a = h(b) and whenever G ∋ b ◦ is B-generic, σ = σ G , and G0 = G ∩ A, then (a) (b) (c) (d) (e) σ:N ≺N σ(θ, A, B, s) = θ, A, B, s CδN (rng(σ)) = CδN (rng(σ0 )) (δ = δ(B)) σ ′′ G ⊂ G σ(tG0 ) = σ0 (tG0 ). ⋆⋆⋆⋆⋆ We now prove a theorem about iterations of length ω. ◦ Theorem 2 Let hBi | i < ωi be s.t. B0 = 2, Bi ⊆ Bi+1 and Bi (B̌i+1 /G is subcomplete) for i < ω. Let Bω be the inverse limit of hBi | i < ωi. Then Bω is subcomplete. ◦ Proof . Let θ be big enough that Bi θ̌ verifies the subcompleteness of B̌i+1 /G for − i < ω. Let N = LA τ s.t. Hθ ⊂ N , θ < τ , and N is a ZFC model. Let σ : N ≺ N s.t. σ(θ, hBi | i ≤ ωi, s) = θ, hBi | i ≤ ωi, s, and N is countable and full. Let G = Gω be Bω -generic over N and set Gi = G ∩ Bi . Then Gi is Bi -generic over N . We claim that there is b ∈ Bω \ {0} s.t. whenever G ∋ b is Bω -generic, there is σ ′ ∈ V[G] s.t. (a) (b) (c) (d) σ′ : N ≺ N σ ′ (θ, hBi | i < ωi, s) = θ, hBi | i < ωi, s CδN (rng(σ ′ )) = CδN (rng(σ)), where δ = δ(Bω ). σ ′ ′′ G ⊂ G. ◦ We first construct a sequence bi , σ i (i < ω) s.t. bi ∈ Bi , hi (bi+1 ) = bi (where hi = hBi ,Bi+1 and whenever Gi ∋ bi is Bi -generic, then, letting σi = σiGi , we have: (a′ ) (b′ ) (c′ ) (d′ ) σi : N ≺ N σi (θ, hBi | i ≤ ωi, s) = θ, hBi | i ≤ ωi, s CδN (rng(σi )) = CδN (rng(σ)) σi′′ G ⊂ Gi . July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 55 Iterating subcomplete forcing Now let hxi | i < ωi enumerate N . Set: ui = the N -least u s.t. σ(xi ) ∈ σi (u) and S u ≤ δ =Df δ(B) in N . (This exists, since rng(σ) ⊂ CδN (rng(σi )) = {σi (u) | u ≤ δ in N } by Chapter 3, Lemma 5.5.) σi will satisfy the additional requirements: (e′ ) σ0 = σ ◦ (f′ ) σi (xh ) = σh (xh ) for h < i, where σh =Df σ h (Gi ∩ Bh ). ◦ Bh i (Note Then σh = σ G ⊆ VBi (i.e. the identity is the h , since we assume: V Gi ∩Bh Bh Bi = tGi for t ∈ VBh , h < i.) natural injection of V into V ). Thus t (g′ ) σi (uh ) = σh (uh ) for h < i. ◦ ◦ ◦ ◦ i for a u i ∈ VBi . We set: b0 = 1, σ 0 = σ̌. Given bi , σ i , Note that ui = u G i ◦ Lemma 1.1 then gives us bi+1 , σ i+1 . (Take σi+1 (tGi ) = σi (tGi ) where Bi t = ◦ ◦ hx̌0 , . . . , x̌i , u 0 , . . . , u i i.) Since hi (bi+1 ) = bi , the sequence ~b = hbi | i < ωi is a T thread in hBi | i < ωi. Hence b = bi 6= 0 in Bω , since Bω is the inverse limit. Now i ◦ ◦ Gi ′ ′ let G ∋ b be Bω -generic. Set Gi = G ∩ Bi , σi = σ G i = σ i . Then (a )–(g ) hold for i < ω. By (f′ ) we can define σ ′ : N ≺ N by: σ ′ (x) = σi (x) for i s.t. σi (x) = σj (x) for i ≤ j. (a), (b) are then trivial. We prove: (c) CδN (rng(σ ′ )) = CδN (rng(σ)). Proof . Set Ci = CδN (rng(σi )) for i < ω. (Hence C0 = CδN (rng(σ)).) (⊂) It suffices to show rng(σ ′ ) ⊂ C0 . But σ ′ (xi ) = σi (xi ) ∈ Ci = C0 . (⊃) We show rng(σ) ⊂ CδN (rng(σ ′ )). S σ(xi ) ∈ σi (ui ) = σ ′ (ui ) ⊂ {σ ′ (u) | u ≤ δ in N } = CδN (rng(σ ′ )). QED(c) Finally we show: (d) σ ′ ′′ G ⊂ G. Proof . We first note that σ ′ ′′ Gi ⊂ G for i < ω, since if a ∈ Gi , then σ ′ (a) = σj (a) ∈ Gj ⊂ G for some j ≥ i. Now let a ∈ G. Since Bω is the inverse limit of T ai where ha | i < ωi is a thread hBi | i < ωi, we may assume w.l.o.g. that a = i<ω in hBi | i < ωi. Let σ ′ (hai | i < ωi) = hai | i < ωi. Then hai | i < ωi is a thread T T in hBi | i < ωi and σ ′ (a) = σ ′ ( ai ) = ai ∈ G by the completeness of G wrt. V, i since ai ∈ G for i < ω. i QED(Theorem 2) Note Theorem 2 can be generalized to countable support iterations of length < ω2 . At ω2 it can fail, however, since in a countable support iteration we are required to take a direct limit at ω2 . If some earlier stage changed the cofinality of ω2 to ω (e.g. if B1 were Namba forcing), then the direct limit would not be subcomplete. Hence for longer iterations we must employ revised countable support iterations, which we discuss in the next section. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 56 Subcomplete Forcing and L-Forcing Revised countable support iterations Definition By an iteration we mean a sequence hBi | i < αi s.t. • B0 = 2 • Bi ⊆ Bj for i ≤ j < α S • If λ < α is a limit ordinal, then Bi generates Bλ . i<λ In dealing with an iteration we shall employ obvious notational simplifications, writing e.g. i for Bi , [[ϕ]]i for [[ϕ]]Bi etc. We also write: hi (b) = hBi (b) =Df T S {a ∈ Bi | b ⊂ a} in Bi , for b ∈ Bj . Recall that: j<α • hi (b) 6= 0 ↔ b 6= 0 S S hi (bj ) • hi ( b j ) = i∈I j∈I • a ∩ hi (b) = hi (a ∩ b) for a ∈ Bi ◦ • hi (b) = [[b̌/G 6= 0]]i . Our definition of “iteration” permits great leeway in defining Bλ at limit λ. In practice people usually employ one of a number of standard limiting procedures, such as finite support (FS), countable support (CS) or revised countable support (RCS) iterations. RCS iterations are particulary suited to subcomplete forcing. The definition of RCS iteration is given in Chapter 0. For present purposes all we need to know is: Fact Let B = hBi | i < αi be an RSC iteration. Then: (a) If λ < α and hξi | i < ωi is monotone and cofinal in λ, then: T bi 6= ∅ in Bi . (i) If hbi | i < ωi is a thread through hBξi | i < ωi, then i<ω T (ii) The set of such bi is dense in Bλ . i (b) If λ < α and i cf(λ̌) > ω for i < λ, then S Bi is dense in Bλ . i<λ (c) If i < λ and G is Bi -generic, then the iteration hBi+j /G | j < α − ii satisfies (a), (b) in V[G]. (Note By a “thread” through hBi | i < ωi we mean a sequence hbi | i < ωi wrt. b0 6= 0, bi ∈ Bi , and hi (bj ) = bi for i ≤ j < ω.) Theorem 3 Let B = hBi | i < αi be an RCS-iteration s.t. for all i + 1 < α: (a) Bi 6= Bi+1 (b) (c) ◦ (B̌i+1 /G is subcomplete) i+1 (δ(B̌i ) has cardinality ≤ ω1 ). i Then every Bi is subcomplete. jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 57 Iterating subcomplete forcing Proof . Set: δi = δ(Bi ). Then (1) δi ≤ δj for i ≤ j < α, since if X is dense in Bj , then {hi (a) | a ∈ X} is dense in Bi . (2) ν ≤ δν for ν < α. Proof . Suppose not. Let ν be the least counterexample. Then ν > 0 is a cardinal. If ν < ω, then δν < ω and hence Bν is atomic with δν the number of atoms. Let ν = n + 1. Then δn < δν < n + 1 by (a). Hence δn < n. Contradiction! Hence ν ≥ ω is a cardinal. If ν is a limit cardinal, then δν ≥ sup δν ≥ ν. Contradiction! i<ν Thus ν is a successor cardinal. Let X ⊂ Bν be dense in Bν with X = δν < ν. Then X ⊂ Bη for an η < ν by the regularity of ν. Hence Bη = Bν , contradicting (a). QED(2) By induction on i we prove: Claim Let G be Bh -generic, h ≤ i. Then Bi /G is subcomplete in V[G]. (Hence Bi ≃ Bi /{1} is subcomplete in V, taking h = 0.) The case h = i is trivial, since then Bi /G ≃ 2. Hence i = 0 is trivial. Now let i = j + 1. Then Bj /G ⊂ Bi /G. Let G̃ be Bj /G-generic over V[G]. Then G′ = G ∗ G̃ = {b ∈ Bj | b/G ∈ G̃} is Bj -generic over V. But then (Bi /G)/G̃ ≃ Bi /G′ is subcomplete in V[G′ ] = V[G][G̃] by (b). ◦ Hence we have shown: ((Biˇ/G)/G is subcomplete).But Bj /G is subcomplete Bj /G in V[G] by the induction hypothesis, so it follows by the two step theorem that Bi /G is subcomplete in V[G]. There remains the case that i = λ is a limit ordinal. By our induction hypothesis Bj /Gh is subcomplete in V[Gh ] for h ≤ j < λ. But then hBh+i /Gh | i < λ − hi satisfies the same induction hypothesis, since if i ≤ k < λ − h and Ĝ is Bh+i /Gh -generic over V[Gh ], then G = Gh ∗ G̃ is Bh+i -generic over V and (Bh+k /Gh )/G̃ ≃ Bh+k /G is subcomplete in V. Case 1 cf(λ) ≤ δi for an i < λ. Then cf(λ) ≤ ω1 in V[Gj ] for i < j < λ whenever Gj is Bj -generic. It suffices to prove the claim for such j, since if h < j and Gh is Bh -generic, we can then use the two step theorem to show – exactly as in the successor case – that Bλ /Gh is subcomplete in V[Gh ]. Hence it will suffice to prove: Claim Assume cf(λ) ≤ ω1 in V. Then Bλ is subcomplete, since the same proof can then be carried out in V[Gj ] to show that Bλ /Gj is subcomplete. Fix f : ω1 → λ s.t. sup f ′′ ω1 = λ. Let θ > λ be a cardinal s.t. B < θ and θ is big enough that: ◦ i (θ̌ witnesses the subcompleteness of B̌j /G) − for i ≤ j < λ. Let N = LA τ be a ZFC model s.t. Hθ ⊂ N , θ < τ . Let σ : N ≺ N s.t. N is countable and full. Suppose also that: σ(θ, B, λ, f , s) = θ, B, λ, f, s. Claim There is b ∈ Bλ \ {0} s.t. whenever G ∋ b is Bλ -generic, there is σ ′ ∈ V[G] s.t. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 58 (a) (b) (c) (d) jensen Subcomplete Forcing and L-Forcing σ′ : N ≺ N σ ′ (θ, B, λ, f , s) = θ, B, λ, f, s CδN (rng(σ ′ )) = CδN (rng(σ)), where σ = sup{δi | i < λ}. σ ′ ′′ G ⊂ G. Set: λ̃ = sup σ ′′ λ. It is easily verified that there is a sequence hνi | i < ωi in ω1N s.t., setting ξ i = f (νi ), we have: ξ 0 = 0, and hξ | i < ωi is monotone and cofinal in λ. (We can assume w.l.o.g. that f (0) = 0.) Set ξi = f (νi ). Then ξi = σ(ξ i ) and hξi | i < ωi is monotone and cofinal in λ̃. Moreover: (3) σ ′ (ξ i ) = ξi whenever σ ′ : N ≺ N s.t. σ ′ (f ) = f . ◦ We now closely imitate the proof of Theorem 2, constructing a sequence bi , σ i (i < ω) s.t. bi ∈ Bξi , hξi (bi+1 ) = bi , and whenever Gi ∋ bi is Bξi -generic, then, ◦ i letting σi = σ G i , we have: (a′ ) (b′ ) (c′ ) (d′ ) (e′ ) (f′ ) σi : N ≺ N σi (θ, B, f , s) = θ, B, f, s CδNi (rng(σi )) = CδN (rng(σ)) σi ′′ Gi ⊂ Gi σ0 = σ ◦ i σi (xh ) = σh (xh ) (h ≤ i) where σh = σ G h (hxℓ | ℓ < ωi being an arbitrarily chosen enumeration of N .) (g′ ) σi (uh ) = σh (uh ) (h ≤ i), where ui = the N -least u s.t. σ(xi ) ∈ σi (u) and u < δ =Df σ −1 (δ) in N . The construction is exactly as before using that σi (Bξj ) = Bξj for all j and that ◦ (B̌ξj+1 /G is subcomplete). As before set: σ ′ (x) = σi (x), where i is big enough that σi (x) = σj (x) for i ≤ j. The verification of (a)–(c) is exactly as before. To verify (d), we first note that, as before, Bξj (2) σ ′ ′′ Gi ⊂ G for i < ω. We then consider two cases: If cf(λ) = ω in N , then cf(λ) = ω in N and λ̃ = λ. Bλ is then the inverse limit of hBξi | i < ωi and Bλ is the inverse limit of hBξi | i < ωi. S Bi We then proceed exactly as before. If cf(λ) = ω1 , Bλ is the direct limit – i.e. i<ω is dense in Bλ . The conclusion then follows by (2). QED(Case 1) Case 2 Case 1 fails. Then λ is regular and δi < λ for all i < λ. Hence λ = sup δi . Let N , N , θ, σ, G be as i<λ before with σ(θ, B, s, λ) = θ, B, s, λ. (However, there is now nothing corresponding to the function f .) As before set: λ̃ = sup σ ′′ λ. It suffices to show: Claim There is c ∈ Bλ s.t. whenever G ∋ c is Bλ -generic, there is σ ′ ∈ V[G] with: (a) σ ′ : N ≺ N (b) σ ′ (θ, B, λ, s) = θ, B, λ, s July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 59 Iterating subcomplete forcing (c) CλN (rng(σ ′ )) = CλN (rng(σ)) (d) σ ′ ′′ G ⊂ G. Choose a sequence hξ i | i < ωi which is monotone and cofinal in λ with ξ 0 = 0. ◦ Set: ξi = σ(ξ i ). As before, our strategy is to construct ci , σ i (i < ω) s.t. c0 = 1, ◦ ◦ ˇ ≺ Ň . The σ 0 = σ̌, hci | i < ωi is a thread in hBξi | i < ωi, and ci forces σ i : N T intention is, again, that if c = ci ∈ G and G is Bλ -generic, then we can define the i ◦ embedding σ ′ ∈ V[G] from the sequence σi = σ G i (i < ω). However, since we no longer have the function f available in defining hξi | i < ωi, we shall not be able to enforce: σi (ξ j ) = ξj for j < ω. Nonetheless we can enforce: sup σi ′′ λ = λ̃, and shall ◦ have to make do with that. We inductively construct ci ∈ Bξi , σ i ∈ VBξi with the properties: (I) (a) c0 = 1 (b) hξh (ci ) = ch for i = h + 1. (II) Let G ∋ ci be Bξi -generic. Set: Gη = G ∩ Bη (η ≤ ξi ), Gη = G ∩ Bη (η ≤ ξ i ), ◦ ◦G ξ h for h ≤ i. Then: σh = σ G h = σh σi : N ≺ N σi (θ, B, λ, s) = θ, B, λ, s CλN (rng(σi )) = CλN (rng(σ)) Let σi (ξ m ) ≤ ξi < σi (ξ m+1 ). Then σi ′′ Gξ ⊂ G. m σi (xh ) = σh (xh ) for h < i, hxℓ | ℓ < ωi being a fixed enumeration of N . σi (uh ) = σh (uh ) for h < i, where uh = the N -least u s.t. σ(xh ) ∈ σh (u). σi = σh if σh (ξ m ) ≤ ξh < ξi < σh (ξ m+1 ) T (I), (II) are easily seen to imply the claim. Set c = ci . Then c 6= 0, since c is a (a) (b) (c) (d) (e) (f) (g) i ◦ thread in hBξi | i < ωi. Let G ∋ c be Bλ -generic. Define σi = σ G i (i < ω) and define σ ′ (x) = σj (x) where σj (x) = σk (x) for all k ≥ j. (a)–(c) follow exactly as before. We prove (d). Since Bλ is the direct limit of hBξ i | i < ωi, it suffices to show: (d’) σ ′ ′′ G ⊂ G for i < ω, where Gη = G ∩ B η . Proof . Let a ∈ Gξi . We first note that for j ≥ i sufficiently large we have: σj (ξ m ) ≤ ξj < σj (ξ m+1 ) for an m ≥ i, since otherwise ξj < σj (ξ i ) for arbitrarily large j. But σj (ξ i ) = σ ′ (ξ i ) for sufficient large j. Hence σ ′ (ξ i ) ≥ sup ξj = λ. Contradiction! j If we also pick j large enough that σj (a) = σ ′ (a), then σ ′ (a) = σj (a) ∈ G, since QED(d) a ∈ Gξm . ◦ It remains only to construct ci , σ i and verify (I), (II). This will be somewhat trickier than the construction in Theorem 2. We shall also have to add further induction hypotheses to (I), (II). Before defining ci we define a bi ∈ Bξi s.t. ◦ (III) (a) b0 = 1, σ 0 = σ̌ July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 60 Subcomplete Forcing and L-Forcing (b) hξj (bi ) = cj if i = j + 1 (c) (II)(a)–(g) hold whenever bi ∈ G. ◦ σ i will be defined simultaneously with bi , before defining ci . Our next induction hypothesis states an important porperty of bi : Definition Let ν ≤ ξi < µ < λ̃ s.t. ξh < ν for h < i, ◦ ◦ ajνµ =Df bi ∩ [[ σ i (ˇξ j ) = ν̌ ∧ σ i (ˇξ j+1 ) = µ̌]]ξi . It follows easily that: (4) ajνµ ∩ aj ′ ν ′ µ′ =0 hj, ν, µi 6= hj ′ , ν ′ , µ′ i if ′ ′ ′ Proof . Suppose ajνµ ∩ aj ν µ ∈ G where G is Bξi -generic. Then j = j ′ , since if e.g. j < j ′ , then µ ≤ σi (ξ j+1 ) ≤ σi (ξ j ′ ) = ν ′ ≤ ξi Contradiction! But then QED(1) ν = σi (ξj ) = ν ′ , µ = σi (ξ j+1 ) = µ′ . Contradiction! Our final induction hypothesis reads: ◦ (IV) ajνµ ∩ [[ σ i (x̌) = y̌]]ξi ∈ Bν if sup ξh < ν ≤ ξi < µ. h<i Hence ajνµ = ajνµ ∩ ◦ [[ σ i (0̌) = 0̌]] ∈ Bν . Definition A = Ai = the set of all ajνµ 6= 0 s.t. sup ξh < ν ≤ ξi < µ. h<i ◦ By (IV) we see that for each a = ajνµ ∈ A there is σ a ∈ VBν s.t. ◦ G ν (5) σ G a = σi for Bξi -generic G ∋ a. But: (6) If G ∋ a is Bν -generic, then G extends to a Bξi -generic G′ s.t. G = G′ ∩ Bν . ◦ ◦G ′ ◦G ′ Hence: σ G a = σa = σi . Thus we have: (7) Let G ∋ a be Bν -generic, where a = ajνµ ∈ Ai . Then (II) holds with ◦ ◦G ◦G ξ h σa = σ G for h < i, where Gη =Df G ∩ Bη a in place of σi , σh = σ h = σ h (η ≤ ν). Note Since a ∈ G, (d) then reduces to: σa ′′ Gξj ⊂ G. Note We then have: σa (xh ) = σh (xh ), σa (uh ) = σh (uh ) for h < i. Whenever ν < µ < λ and G is Bν -generic, we know that Bµ /G is subcomplete in ◦ V[G]. Then, using (7), Lemma 1.1, and repeating the construction of bi+1 , σ i+1 ◦ from bi , σ i in the proof of Theorem 2, we get: jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Iterating subcomplete forcing jensen 61 ◦ (8) Let a ∈ Ai , a = ajνµ . There are ã ∈ Bµ , σ ′a ∈ VBµ s.t. hν (ã) = a and ◦ ◦ whenever G ∋ ã is Bµ -generic, σa = σ G , and σa′ = σ ′G a , then we have: σa′ : N ≺ N τa′ (θ, B, λ, s) = θ, B, λ, s CλN (rng(σa′ )) = CλN (rng(σa )) σa′ ′′ Gξj+1 ⊂ G Let r be least s.t. µ ≤ ξr . Then σa′ (xh ) = σa (xh ) for h < r. Let r be as above. Then σa′ (uah ) = σa (uah ) for h < r, where uah = the N -least u ∈ N s.t. u ≤ λ in N and σ(xh ) ∈ σa (uh ). (g) σa′ (ξ ℓ ) = σa (ξ ℓ ) for ℓ ≤ j + 1. (a) (b) (c) (d) (e) (f) ◦ For each a ∈ Ai fix such a pair ã, σ ′a , which can be regarded as an instruction to be used later in forming br , where r is least s.t. µ ≤ ξr . If G is Bξr -generic and ◦ ◦G a ∩ br ∈ G, we shall want: ã ∈ G and σr = σ ′G a (where σr = σ r ). In particular, we want: a ∩ br = ã. But we shall also require: hξi (br ) = ci . Hence we need: a ∩ ci = hξ (a ∩ br ) = hξ (ã). This is why bi must be “shrunk” to ci . Accordingly we define ci as follows: S Definition Let bi be given. Set b = bi \ Ai . Then: [ ci =Df b ∪ hξi (ã). a∈Ai We are working by induction on i. We assume (I)–(IV) to hold below i and (III), (IV) to hold at i. We must now verify (I), (II) at i. (II) is immediate by (III)(c), since ci ⊂ bi . (I)(b) holds, since hξj hξi (ã) = hξj (ã) = hξj hν (ã) = hξj (a) for i = j +1 and a = aℓµν ∈ Ai . Hence: [ [ hξj (ã) = hξj (b ∪ hξi (ã)) = hξj (bi ) = cj . hξj (ci ) = hξj (b) ∪ a∈A a∈A For (I)(a) note that A0 = {a} where a = a0,0,ξ1 = 1, since σ0 = σ by (III)(a). Hence c0 = h0 (ã) = 1. This completes the verification of (I)–(IV) at i, given (III), (IV) at ◦ i and (I)–(IV) below i. Now let (I)–(IV) hold below i. We must define bi , σ i and ◦ verify (III), (IV) at i. For i = 0 set: b0 = 1, σ 0 = σ̌. The verifications are trivial. Now let i = j + 1. Note that Aℓ , hã | a ∈ Aℓ i, hσa′ | a ∈ Aℓ i have been defined for ℓ ≤ j. Set: S Aℓ s.t. ξj < µ. Definition Âj = the set of a = ahνµ ∈ ℓ≤j ′ (9) Let a, a′ ∈ Âj , a = ahνµ , a′ = ah ν ′ ′ µ . Then a ∩ a′ = 0 if hh, ν, µi = 6 hh′ , ν ′ , µ′ i. Proof . Suppose not. Let a ∈ Aℓ , a′ ∈ Aℓ′ . Then ℓ 6= ℓ′ by (4). Let e.g. ℓ < ℓ′ Let ◦ a ∩ a′ ∈ G, where G is Bj -generic. Set σℓ = σ G ℓ for ℓ ≤ j. Then: σℓ (ξ h ) = ν ≤ ξℓ < ν ′ ≤ ξℓ′ ≤ ξj < µ = σℓ (ξ h+1 ). July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 62 jensen Subcomplete Forcing and L-Forcing Hence σℓ′ = σℓ by (II)(g). Hence h < h′ , since σℓ (ξ h′ ) = ν ′ > σℓ (ξ h ). Hence σℓ (ξ h+1 ) ≤ ν ′ < µ. Contradiction! QED(9) We now define: Definition bi = S {hξi (ã) | a ∈ Âj } for i = j + 1. ◦ To define σi we set: à = the set of aiνµ ∈ Âj s.t. µ ≤ ξi . σ i ∈ VBi is then a name S ◦ ◦ ◦ ◦ s.t. [[ σ i = σ ′a ]] = ã if a ∈ Ã, [[ σ i = σ j ]] ∩ bi = bi \ Ã. Then: (10) (III)(c) holds at i. Proof . Let G ∋ bi be Bξi -generic. Case 1 ã ∈ G for an a ∈ Ã. Let a = ahνµ ∈ Aℓ , µ ≤ ξi (hence ξj < µ ≤ ξi ). Thus σi = σa′ . (II)(a)–(d) hold by (8)(a)–(d). Note that the r in (8)(e), (f) is r = i. But, if a ∈ Aℓ , ℓ ≤ j, then σℓ = σa . Hence σℓ (ξ h ) = ν ≤ ξℓ ≤ ξℓ′ < σℓ (ξ h+1 ) = µ for ℓ ≤ ℓ′ ≤ j. Hence: σa = σℓ′ for ℓ ≤ ℓ′ ≤ j. But then (II)(e), (f) hold by (8)(e), (f). Finally (II)(g) holds vacuously, since ξj < µ = σi (ξ h+1 ) ≤ ξi , hence ξj < σi (ξ m ) where σi (ξ m ) ≤ ξi < σi (ξ m+1 ). Case 2 Case 1 fails. Then σi = σj . (II)(a)–(g) then follow trivially. QED(10) (III)(a) holds vacuously at i = j + 1. We prove: (11) (III)(b) holds at i. Proof . Clearly hξj (bi ) = S hξj (ã). Hence we need: a∈Âj Claim cj = S hξj (ã). a∈Âj S For j = 0 this is trivial, so let j = ℓ + 1. Recall that cj = b ∪ hξj (ã), where a∈Aj S a, so it suffices to show: b = bj \ a∈Aj Claim b = S hξj (ã) where A′ = Âj \ Aj . a∈A′ ′ (⊃) Let a′ ∈ A . Then a′ ∈ Âℓ . Hence hξj (ã′ ) ⊂ S a∈Âℓ ′ hj (ã) = bj . But for all a ∈ Aj we have a ∩ a′ = 0 by (10). Hence hξj (ã) ∩ hξj (ã ) = a ∩ hξj (ã′ ) = hξj (a ∩ ã′ ) = 0, since a ∩ ã′ ⊂ a ∩ a′ = 0. Hence hξj (ã′ ) ⊂ b. (⊂) Suppose not. Then there is a ∈ Âj \ A′ s.t. b ∩ hξj (ã) 6= 0. But then a ∈ Aj QED(11) and hξj (ã) = a. Hence a ∩ b = 0 by the definition of b. It remains only to show: (12) (IV) holds at i. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 63 Iterating subcomplete forcing ◦ Proof . Let a = ah,ν,µ ∈ Ai . Then ξj < ν ≤ ξi . a ∩ [[ σ i (x̌) = y̌]] = bi ∩ d, where ◦ ◦ ◦ ◦ d = [[ σ i (ˇξ h ) = ν̌ ∧ σ i (ξ h+1 ) = µ̌ ∧ σ i (x̌) = y̌]]Bξi = [[ϕ( σ i )]], where the formula , ν̌, µ̌, x̌, y̌, all of which lie in V2 . Recall that ϕ(v) is Σ in the parameters ˇξ , ˇξ 0 h h+1 we are assuming VBη ⊆ VBτ for η ≤ τ (i.e. Bη is completely contained in Bτ and the identity is the natural embedding of VBη in VBτ ). As mentioned in Chapter 0, this has the consequence that if ψ is a Σ0 formula and t1 , . . . , tm ∈ VBη , then: a Bτ ψ(~t ) ←→ a Bη ψ(~t ) for a ∈ Bη , or in other words: [[ψ(~t )]]Bτ = [[ψ(~t )]]Bη ∈ Bη . S hξi (ẽ). Hence it suffices to We shall make strong use of this. We know: bi = e∈Âj ∗ assign to each e ∈ Âj an e ∈ Bν s.t. hξi (ẽ) ∩ d = e∗ , since then we have: bi ∩ d = [ e∗ ∈ Bν . e∈Âj For hξi (ẽ) ∩ d = 0 we, of course, set e∗ = 0. Now let hξi (ẽ) ∩ d 6= 0. Let e = ah,ν,µ ∈ ◦ ◦ ◦ Âj . Let G ∋ hξi (ẽ) ∩ d be Bξi -generic. Set: σi = σ G , σ j = σ G j . Case 1 µ ≤ ξi . ◦ ◦ Then ẽ = hξi (ẽ) ∈ Bµ ∧ G. Hence σi = σe′ =Df σ ′e G . Hence [[ϕ( σ ′e )]] ∈ G. ◦ Conversely, if ẽ ∩ [[ϕ( σ ′e )]] ∈ G, then σi = σe′ and hence ẽ ∩ d ∈ G. Since this holds for all G, we conclude: ◦ ẽ ∩ d = ẽ ∩ [[ϕ( σ ′e )]] ∈ Bµ . However, µ ≤ ν, since otherwise we would have σi (ξ ℓ ) = σj (ξ ℓ ) for ℓ ≤ h + 1 and σi (ξ h ) = ν < µ = σi (ξ h+1 ). Hence h ≤ h and σi (ξ h ) ≤ σj (ξ h ) = ν ≤ ξj < ν. Contradiction! QED(Case 1) Case 2 µ > ξi . We show that this cannot occur. Clearly, if G ∋ hξi (ẽ) ∩ d is Bξi -generic, then σi = σj = σe′ by the definition of σe′ . But then ẽ ∩ d = 0, since if G ∋ ẽ ∩ d were Bµ -generic, then σi (ξ h ) = ν ≤ ξj < ν ≤ ξi < µ = σi (ξ h+1 ). Hence ν = σi (ξ h ) is impossible. Contradiction! Since d ∈ Bξi , we conclude: hξi (ẽ) ∩ d = hξi (ẽ ∩ d) = 0. Contradiction. QED(12) This completes the proof of Theorem 3. The above theorem can be adapted to iterations which allow more freedom in the formation of limit algebras. Definition An iteration B = hBi | i < αi is nicely subcomplete iff the following hold: July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 64 jensen Subcomplete Forcing and L-Forcing (a) For all i + 1 < α: ◦ B̌i+1 /G is subcomplete, i+1 card(δ(B̌i )) ≤ ω1 . (i) (ii) i (b) If λ < α and hξn | n < ωi is monotone and cofinal in λ, then T (i) bn 6= 0 in Bλ whenever b = hbn | n < ωi is a thread in hBξn | n < ωi, n (ii) Bλ is subcomplete if Bi is subcomplete for i < λ. S Bi is dense in Bλ . (c) If λ < α and i cf(λ̌) > ω for all i < λ, then i<λ (d) If i < α and G is Bi -generic, then (a)–(c) hold for hBi+j /G | j < α − ii in V[G]. (This allows greater freedom in forming limit algebras at points which acquire cofinality ω, but requires us to take direct limits at other points.) Theorem 4 subcomplete. Let B = hBi | i < αi be nicely subcomplete. Then every Bi is Proof . (sketch) By induction on i we again prove: Claim Let h ≤ i. Let G be Bh -generic. Then Bi /G is subcomplete in V[G]. The cases h = i, i = j + 1 are again trivial, so assume that i = λ is a limit ordinal. We again have the two cases: Case 1 cf(λ) ≤ δ(Bh ) for an h < λ. Case 2 Case 1 fails. In Case 1 it again suffices to prove the claim for sufficiently large h < λ, so we assume cf(λ) ≤ ω1 in V[G] whenever G is Bh -generic. But then we can assume cf(λ) ≤ ω1 in V, since the same proof can be carried out in V[G] for hBh+j /G | j < α − hi. This splits into two subcases: Case 1.1 cf(λ) = ω. Then Bλ is subcomplete by (b)(ii). Case 1.2 cf(λ) = ω1 . We then literally repeat the argument in the proof of Theorem 3, using that Bλ is the direct limit of hBi | i < λi. (Note If we instead assumed cf(λ) = ω, the proof in Theorem 3 would no longer T work, since the set of b s.t. b = hbn | n < ωi is a thread in hBξn | n < ωi may not n be dense in Bλ .) QED(Case 1) In Case 2 we literally repeat the proof in Theorem 3, using that if λ̃ = sup σ ′′ λ, then T by (b)(i), if c = cn is a thread in hBξn | n < ωi (ξn = σ(ξ n ), where hξ n | n < ωi n is monotone and cofinal in λ), then c ∈ Bλ̃ . Just as before we utilize the fact that we can ensure that sup σn ′′ λ = λ, even though we cannot fix the values of σn (ξ i ) (i < ω). QED(Theorem 4) July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 65 Iterating subcomplete forcing Forcing Axioms We say that a complete BA B satifies Martin’s Axiom iff whenever h∆i | i < ω1 i is a sequence of dense sets in B, there is a filter G on B s.t. G ∩ ∆i 6= ∅ for i < ω. The original Martin’s Axiom said that this holds for all B satisfying the countable chain condition. This axiom is consistent relative to ZFC. It was later discovered that very strong versions of Martin’s Axiom can be proven consistent relative to a supercompact cardinal. The best known of these are the proper forcing axiom (PFA), which posits Martin’s Axiom for proper forcings and Martin’s Maximum (MM) which is equivalent to Martin’s Axiom for semiproper forcings. Both of these strengthen the original Martin’s Axiom, hence imply the negation of CH. Here we shall consider the subcomplete forcing axiom (SCFA), which says that Martin’s Axiom holds for subcomplete forcings. This, it turns out, is compatible with CH, hence cannot be a strengthening of the original Martin’s Axiom (though it is, of course, a strengthening of Martin’s Axiom for complete forcings). Nonetheless it turns out that SCFA has some of the more striking consequences of MM. A fuller account of this can be found in [FA]. We recall from Chapter 3.1 that the notion of subcompleteness is “locally based” in the sense that, if θ, θ′ are cardinals with H θ < θ′ , then we need only consider ′ N = LA τ of size less than θ , in order to determine whether θ verifies the subcompleteness of a given B. In other words, P(Hθ ) contains all the information needed to determine this. As a consequence we get Chapter 3, Corollary 2.3, which says that, if W is an inner model and P(Hθ ) ⊂ W , then the question, whether θ verifies the subcompleteness of B, is absolute in W . Using this we prove: Theorem 5 Let κ be supercompact. There is a subcomplete B ⊂ Vκ s.t. whenever G is B-generic, then: (a) κ = ω2 , (b) CH holds, (c) SCFA holds. Proof . Let f be a Laver function (i.e. for each x and each cardinal β there is a supercompact embedding π : V → W s.t. x = π(f )(κ) and W β ⊂ W ). We define and RCS iteration hBi | i ≤ κi by: • B0 = 2. ◦ • If i f (i) is a subcomplete forcing, then i B̌i+1 /G ≃ f (i) ∗ coll(w1 , f (i)). • If 6 i f (i) is a subcomplete forcing, then i B̌i+1 /G ≃ coll(w1 , w2 ). ◦ Let G be B = Bκ -generic. Then CH holds in V[G], since there will be a stage Bi+1 which makes CH true by collapsing. But then it remains true at later stages, since no reals are added. We now show that SCFA holds in V[G]. Let A ∈ V[G] be July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 66 Subcomplete Forcing and L-Forcing ◦ subcomplete. Let A = AG . Let U ∈ V, U ⊂ VBκ s.t. ◦ (1) [[x ∈ A]] ⊂ S [[x = z]] for x ∈ VBκ . z∈U ◦ We may also assume w.l.o.g. that A is forced to be subcomplete and in fact: ◦ (2) κ θ̌ verifies the subcompleteness of A. ◦ Let β = Vβ where A, U, θ ∈ Vβ . Let π : V → W be a supercompact embedding ◦ s.t. A = π(f )(κ) and W β ⊂ W . (Hence Vβ+1 ⊂ W .) Then: (3) θ verifies the subcompleteness of A in W [G], since this depends only on P(Hθ ) ⊂ W . Now let: hB′i | i ≤ κ′ i = π(hBi | i ≤ κi). Then Bκ = B′κ and G is B′κ -generic over W . Hence we can form G′ ⊃ G which is B′κ -generic over W . Since B′κ+1 ≃ A ∗ coll(ω1 , A), there is A ∈ W [G′ ] which is A-generic over W [G]. Now let π ∗ be the unique π ∗ ⊃ π s.t. (4) π ∗ : V[G] ≺ W [G′ ] ∧ π ∗ (G) = G′ . Then ◦ ′ (5) π ∗ (A) = A′ , where A′ = π(A)Bκ′ . But (6) π ∗ ↾ A ∈ W [G′ ], since π ↾ U ∈ W and π ∗ ↾ A is definable as that π̃ s.t. ′ π̃(tG ) = π(t)G whenever t ∈ U and tG ∈ A. Let A′ be the filter on A′ generated by π ∗ ′′ A. Let h∆i | i < ω1 i be a sequence of dense sets in A in V[G]. Let h∆′i | i < ω1 i = π ∗ (h∆i | i < ω1 i). Obviously A′ ∩ ∆′i 6= ∅ for i < ω1 . Since π : V[G] ≺ W [G], we conclude that there is a filter à on A in V[G] s.t. à ∩ ∆i 6= ∅ for all i < ω2 . QED(Theorem 5) In [FA] we show that subcomplete forcings are ♦-preserving – i.e. if ♦ holds in V, it continues to hold in V[G]. It follows easily from this that ♦ holds in the model V[G] just constructed. If we do a prior application of Silver forcing to make GCH true, then the ultimate model will also satisfy GCH. Hence we have, in fact, shown the consistency of SCFA + ♦ + GCH relative to a supercompact cardinal. SCFA has two of the more striking consequences of MM: Friedman’s principle and the singular cardinal hypothesis at singular strong limit cardinals. Friedman’s principle at a regular cardinal τ > ω1 says that if A ⊂ τ is any stationary set of ω-cofinal ordinals, then there is a normal function f : ω1 → A (i.e. f is monotone jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 67 Iterating subcomplete forcing and continuous at limits). It is easily seen that Friedman’s principle at β + implies the negation of β . Lemma 6 Assume SCFA. Let κ > ω1 be regular. Then Friedman’s principle holds at κ. Proof . Let PA be as in the final example of Chapter 3.3, where A ⊂ κ is a stationary set of ω-cofinal ordinals. Let ∆i = the set of p ∈ PA s.t. i + 1 ⊂ dom(p) for i < ω1 . Then ∆i is dense in PA . By SCFA there is a set G of mutually compatible conditions S s.t. G ∩ ∆i = ∅ for i < ω1 . But then the function f = G has the desired property. QED(Lemma 6) By essentially the same proof we get. Lemma 7.1 Assume SCFA. Let τ > ω1 be regular. Let Ai ⊂ τ be a stationary set of ω-cofinal points for i < ω1 . Let hDi | i < ω1 i be a partition of ω1 into disjoint stationary sets. Then there is a normal function f : ω1 → τ s.t. f (j) ∈ Ai for j ∈ Di . Proof . We need only to show that the appropriate forcing P is subcomplete The proof is exactly like Chapter 3.3, Lemma 6.3. QED(Lemma 7.1) The singular cardinal hypothesis for strong limit cardinals then follows by a well known argument of Solovay: Corollary 7.2 Assume SCFA. Let τ be as above. Then τ ω1 = τ . Proof . Let hAξ | ξ < τ i partition {λ < τ | cf(λ) = ω} into disjoint stationary sets. S For each a ∈ [τ ]ω1 let hξi | i < ω1 i enumerate a. Let f : ω1 → Aξi be normal i<ω1 s.t. f (j) ∈ Aξi if j ∈ Di , where hDi | i < ω1 i partitions ω1 into stationary sets. Let λ = sup f ′′ ω1 . Then a = Bλ =Df {ξ | Aξ ∩ λ is stationary in λ}. Hence [τ ]ω1 ⊂ {Bλ | λ < τ }. QED(Corollary 7.2) β Corollary 7.3 Assume SCFA. If cf(β) ≤ ω1 < β and 2 ` ≤ β + , then 2β = β + . β Proof . 2β = (2 ` )cf(β) ≤ (β + )ω1 = β + . Using Silver’s Theorem we conclude: Corollary 7.4 Assume SCFA. If β is a singular strong limit cardinal, then 2β = β+. July 21, 2012 15:2 68 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen Chapter 5 L-Forcing β In the following, assume CH. Let β > ω1 be a cardinal and assume: 2 ` = β (i.e. 2α ≤ β for α < β). Let M = LA β =Df hLβ [A], ∈, A ∩ Lβ [A]i s.t. Lβ [A] = Hβ and A ⊂ Hβ . Suppose we have forcing conditions which do not collapse ω1 , but do add a map collapsing β onto ω1 . The existence of such a map is equivalent to the existence of a commutative “tower” hMi | i < ω1 i, hπij | i ≤ j < ω1 i s.t. each Mi is countable and transitive, πij : Mi → Mj for i ≤ j < ω1 , and the tower converges to M (i.e. there are hπi | i < ωi s.t. Mi hπi | i < ωi is the direct limit of hMi | i < ω1 i, hπij | i ≤ j < ω1 i). In L-forcing we attempt to collapse β onto ω1 by conditions which directly describe such a tower (or at least a commutative directed system converging to M ). The “L” in “L-forcing” refers to an infinitary language on a structure of the form: N = hHβ + , ∈, M, . . .i in the ground model V. L then determines a set of conditions PL . L-forcing has been used to add new reals with interesting properties. In these notes, however, we shall concentrate wholly on a form of L-forcing which does not add new reals. This means, of course, that Hω1 is absolute. Hence all countable initial segments of our “tower” will lie in V. The theory of L-forcing is developed in [LF]. In that paper, however, we dealt only with forcings which literally added a tower converging to M in the aforementioned sense. In later applications we found it better to replace the tower by other sorts of convergence systems. We therefore adopt a more general approach here. The proofs in [LF] can be readily adapted to this approach. Recall that we are working in first order set theory, so we cannot literally quantify over arbitrary classes. Instead we work with “virtual classes”, which are expressions of the form {x | ϕ(x)} where ϕ = ϕ(x) is a formula of ZF. Normally we suppose x to be the only variable occurring free in ϕ. We define: Definition An approximation system is a pair hΓ, Πi of virtual classes s.t. (I)– (VII) below are provable in ZFC− . (I) Γ is a class of pairs hM, Ci s.t. 69 July 21, 2012 15:2 70 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing 1 ,...,An (a) M = LA for some A1 , . . . , An , τ . τ (b) C ⊂ M . (Definition For u ∈ Γ set: u = hMu , Cu i.) (II) Π is a class of triples hπ, u, vi s.t. u, v ∈ Γ, π : Mu ≺ Mv , Cu = π −1 ′′ Cv . W (Definition π : u ⊳ v ↔Df hπ, u, vi ∈ Π, u ⊳ v ↔Df π π : u ⊳ v) (III) There is at most one π s.t. π : u ⊳ v. (Definition πuv ≃Df that π s.t. π : u ⊳ v.) (IV) (a) u ⊳ u ∧ πuu = id for u ∈ Γ. (b) u ⊳ v ⊳ w → (u ⊳ w ∧ πuw = πvw ◦ πuv ). −1 (c) If u, v ⊳ w and rng(πuw ) ⊂ rng(πvw ), then u ⊳ v and πuv = πvw ◦ πuw . We say that a set X ⊂ Γ is ⊳-directed iff for all u, v ∈ X there is w s.t. u, v ⊳ w. In this case we can form a direct limit v, hπu | u ∈ Xi of hu | u ∈ Xi, hπuu′ | u ⊳ u′ ∧ u, u′ ∈ Xi. Then v = hA, Ci, where A is a (possibly ill founded) ZFC− model. If A is well founded, we can take it as transitive. Clearly the transitivized direct limit of X, if it exists, is uniquely determined by X. (V) Let X ⊂ Γ be ⊳-directed. Let v, hπu | u ∈ Xi be the transitivized direct limit. Then v ∈ Γ and πu = πuv for u ∈ X. Moreover, if u ⊳ w for all u ∈ X, then v ⊳ w. (Hence πvw is uniquely determined by: πvw πu = πuw for u ∈ X.) If t = {x | ϕ(x)} is a virtual class and W is any set or class, we can form tW (the interpretation of t in W ) by relativizing all quantifiers in ϕ to W . (VI) If M is an admissible set, then Γ ∩ M = ΓM and Π ∩ M = ΠM . If A = h|A|, ∈A i is any binary structure we can form the relativization tA by relativizing quantifiers to |A| and simultaneously replacing ∈ by ∈A in ϕ. (VIII) If A is a solid model of ZFC− and A = wfc(A), then Γ ∩ A = ΓA ∩ A, and Π ∩ A = ΠA ∩ A. Hence: (1) If V[G] is a generic extension of V, then ΓV[G] ∩ V = ΓV , ΠV[G] ∩ V = ΠV . Proof . Let x ∈ V. Then x ∈ M ∈ V, where M is admissible. Hence x ∈ ΓV[G] ↔ x ∈ ΓM ↔ x ∈ ΓV , applying (VI) first in V[G], then in V. QED(1) Remark In practice (I)–(VII) will follow readily from the definitions given for Γ, Π, so we shall not bother to verify them in detail. In all cases Γ and Π will also be provably primitive recursive in ZFC− , so the absoluteness properties (VI), (VII) will follow by Chapter 1.3. However, it will also be easy to verify these properties directly without going through the theory of pr functions. ⋆⋆⋆⋆⋆ A simple example of an approximation system is: Γ is the set of all hM, Ci s.t. jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 71 L-Forcing • M = LA τ for some A, τ . • M models ZFC− and ω1 exists and CH. • C maps ω1M onto M . Π is then the set of all hπ, u, vi s.t. u, v ∈ Γ, π : Mu ≺ Mv and π ◦ Cu ⊂ Cv . (Note that in this example we have Γ, Π ⊂ Hω2 .) The absoluteness properties are straightforward, since if M, N, π ∈ A and A is admissible, then π : M ≺ N is uniformly expressible by a Σ1 formula in any solid A extending A. Now let an approximation system hΓ, Πi be given. Let M = LA β be as described at the outset with β > ω1 and Hβ = Lβ [A]. Our aim in L-forcing is to generically add C ⊂ M in the extension V[C] s.t. hM, Ci ∈ ΓV[C] and hM, Ci is the limit of a directed X ⊂ Γ ∩ Hω1 . At the same time we want to add no reals, so that Γ ∩ Hω1 remains absolute. Since we are assuming CH it follows easily that card(M ) = ω1 in V[G]. (In the above example we would accomplish this explicitly, since C would map ω1 onto M .) L is a language on N = hHβ + , ∈, M, <, . . .i, where < is a well ordering of Hβ + . (Note N remains a ZFC− model, hence admissible, no matter which predicates and constants we adjoin to it.) The only nonlogical predicate of L is ∈. In addition to the constants x (x ∈ N ) ◦ there will be one further constant C. We always suppose L to contain the following core axioms: • ZFC− (here the usual finite axioms are meant, so we could write them as a single M -finite conjunction). W W V v = z) for x ∈ N . • v(v ∈ x ↔ z∈x • Hω1 = Hω1 (or equivalently P(ω) = P(ω)). ◦ • hM , Ci ∈ Γ. • For all countable X ⊂ M there is u ∈ Γ ∩ Hω1 s.t. X ⊂ rng(π ◦ u,hM , C i ). (L might, of course, contain further axioms as well.) A Definition Let A be a solid model of L. ΓA , ΠA , ⊳A , πuv (u ⊳A v) are defined in the obvious way. Set: ◦ Γ̃ = Γ̃A =Df {e ∈ Γ ∩ Hω1 | e ⊳ hM, C A i in A}. For e ∈ Γ̃ set: πeA =Df π A ◦ e,hM, C A i . ◦ Lemma 1.1 Let A be as above. Then Γ̃ is a ⊳-directed system with limit hM, C A i, hπeA | e ∈ Γ̃i. S rng(πeA ) is trivial. We show that Γ̃ is directed. Let e0 , e1 ∈ Γ̃. Let Proof . M = e∈Γ̃ u ∈ Γ̃ s.t. rng(πeA0 ) ∪ rng(πeA1 ) ⊂ rng(πuA ). Then e0 , e1 ⊳ u and πeh u = (πuA )−1 ◦ πeAh . QED(Lemma 1.1) July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 72 jensen Subcomplete Forcing and L-Forcing Lemma 1.2 Let A be as above. Let A ∈ A s.t. A ⊂ M . There is e ∈ Γ̃ s.t. rng(πeA ) ≺ hM, Ai. S Proof . In A construct hei | i < ωi, hXi | i < ωi s.t. rng(πeAh ) ⊂ Xi ⊂ rng(πeAi ) h<i and Xi ≺ hM, Ai. It follows easily that eh ⊳ ei for h ≤ i < ω and {eh | h < ω} has ◦ S rng(πeAi ) = a direct limit e, hπei e | i < ωi. But then e ⊳ hM, C A i and rng(πeA ) = i<ω S Xi ≺ hM, Ai. QED(Lemma 1.2) i<ω Corollary 1.3 Let A be as above. Let U ⊂ P(M ) s.t. U ∈ A is countable in A. There is e ∈ Γ̃ s.t. rng(πeA ) ≺ hM, Ai for all A ∈ U . Proof . Let hAi | i < ωi ∈ A enumerate U and apply Lemma 1.2 to A = {hx, ii | x ∈ Ai }. QED(Corollary 1.3) Corollary 1.4 Let A be as above. Let U , V be countable in A s.t. U ⊂ M , V ⊂ P(M ). There is e ∈ Γ̃ s.t. U ⊂ rng(πeA ) ≺ hM, Ai for A ∈ V . Proof . Apply Corollary 1.3 to U ∪ V . QED(Corollary 1.4) If L is consistent, we can define a set P = PL of conditions as follows: Definition Let P̃ be the set of p = hp0 , p1 i s.t. p0 ∈ Γ ∩ Hω1 and p1 ⊂ P(M ) × P(Mp0 ) is countable. For p ∈ P̃ let ϕp be the conjunction of the L statements: ◦ • p0 ⊳ hM , Ci • If π = π ◦ p0 ,hM, C i , then π : hM p0 , ai ≺ hM , ai for all ha, ai ∈ p1 . Set L(p) = L + ϕp . We set: P = {p ∈ P̃ | con(L(p))}, where con(L(p)) is the statement that “L(p) is consistent”. The extension relation on P is then defined by: Definition Let p, q ∈ P p ≤ q ←→Df (q0 ⊳ p0 ∧ rng(q1 ) ⊂ rng(p1 ) ∧ πq0 p0 : hMq0 , ai ≺ hMp0 , a′ i whenever ha, ai ∈ q1 , ha, a′ i ∈ p1 ). Lemma 2.1 ≤ is a partial ordering. Proof . Transitivity is immediate. Now let p ≤ q, q ≤ p. We claim that p = q, p0 = q0 is immediate. But if ha, ai ∈ q1 , ha, a′ i ∈ p1 , then a = a′ , since πq0 p0 = id. Hence q1 = p1 . QED(Lemma 2.1) Definition dom(p1 ). Let p ∈ P. Mp = Mp0 , Cp = Cp0 , F p = p1 , Rp = rng(p1 ), Dp = Lemma 2.2 Let p ∈ P. Then July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 73 L-Forcing (a) (F p )−1 is a function. (b) If Rp is closed under set difference, then F p : Dp ↔ Rp . (c) F p ↾ Mp injects Mp into M . Proof . Let A be a solid model of L(p). Let π = πpA =Df (π ◦ p0 ,hM, C A i )A . (a) Let ha, ai, ha, a′ i ∈ F p . Then a = a′ = π −1 ′′ a. (b) Let ha, ai, hb, bi ∈ F p . It suffices to show: Claim: a ⊂ b → a ⊂ b. Set c = a \ b, c = a \ b = ∅. Then (F p )−1 (c) = π −1 ′′ c = π −1 ′′ a \ π −1 ′′ b = b \ a = ∅. QED(b) Hence c = ∅, since π : hMp0 , ci ≺ hM, ci. (c) Let x ∈ Mp0 , hx, xi ∈ F p . Then π(x) = x ∈ M since π : hMp , xi ≺ hM, xi. QED(Lemma 2.2) We define: Definition π p = F p ↾ Mp . Note By the proof of (c) we have: L(p) ⊢ π p ⊂ π ◦ p,hM, C i . We now prove the main lemma on extendability of conditions. Lemma 3.1 P 6= ∅. Moreover, if p, q ∈ P and L(p) ∪ L(q) is consistent, there is r s.t. r ≤ p, q. Moreover, if X ⊂ P(M ) is any countable set, we may choose r s.t. X ⊂ Rr . ◦ Proof . To see P 6= ∅ let A be any solid model of L. Let e ⊳ hM, C A i in A where e ∈ Γ ∩ Hω1 . Then A  L(p) where p = he, ∅i. Hence p ∈ P. Now let A  L(p) ∪ L(q). Let X ⊂ P(M ) be countable in V. Let Y = X ∪ Rp ∪ Rq . ◦ There is e ∈ Hω1 ∩Γ s.t. e⊳hM, C A i in A and πeA ≺ hM, Ai for all A ∈ Y . For A ∈ Y set A = (πeA )−1 ′′ A. Letting hAi | i < ωi be an enumeration of Y in V, we see that hAi | i < ωi ∈ Hω1 . Hence F ∈ V where F = {hA, Ai | A ∈ Y } = {hAi , Ai i | i < ω}. Set r = he, F i. Then A  L(r) and p, q ≤ r. QED(Lemma 3.1) Corollary 3.2 p, q are compatible in P iff L(p) ∪ L(q) is consistent. Proof . (←) Lemma 3.1. (→) If r ≤ p, q, then L(r) ⊢ L(p) ∪ L(q). QED(Corollary 3.2) Corollary 3.3 Let p ∈ P, X ⊂ P(M ) where X is countable. There is r ≤ p with X ⊂ Rr . Corollary 3.4 Let p ∈ P, u ⊂ M , u is countable. There is r ≤ p with u ⊂ rng(π r ). Lemma 3.5 Let p ∈ P, u ⊂ Mp , u finite. There is r ≤ p s.t. r0 = p0 and u ⊂ dom(π r ). Proof . Let A be a solid model of L(p). Set: r0 = p0 , F r = F p ∪ (πpA ↾ u). Then A  L(r). QED(Lemma 3.5) July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 74 Subcomplete Forcing and L-Forcing Using these extension lemmas we get: Lemma 3.6 Let G be P-generic. For p ∈ G set: πpG = Then: S q {π | q ∈ G ∧ p0 = q0 }. (a) {p0 | p ∈ G} is a ⊳-directed system with limit hM, C G i, hπpG | p ∈ Gi, where S C G = πpG ′′ Cp . p (b) πpG : hMp , ai ≺ hM, ai for ha, ai ∈ F p . Note πpG : p0 ⊳ hM, C G i in V[G] by (a). The proof is straightforward. Now let κ > (2β ) be regular in V. Then κ V[G] remains regular in V[G], since P ∈ Hκ . hHκ , C G i then models all of the core axioms except possibly the axiom: Hω1 = Hω1 . We now state a condition called revisability which will guaratee that no reals V[G] are added – hence that all core axioms hold in hHκ , C G i. We first define: Definition Let N ∗ = hHδ , M, <, . . .i be a model of countable or finite type, where δ > 2β is a cardinal and < well orders Hδ . Let p ∈ P. p conforms to N ∗ iff whenever a1 , . . . , an ∈ Rp (n ≥ 0) and b ⊂ M is N ∗ -definable in a1 , . . . , an , then b ∈ Rp . Note If p conforms to N ∗ then Rp 6= ∅ and F p : Dp ↔ Rp by Lemma 2.2. Note {p | p conforms to N ∗ } is dense in P by the extension lemmas. Before defining revisability we prove a theorem: Lemma 4 Let p conforms to N ∗ . There is a unique N ∗ = N ∗ (p, N ∗ ) s.t. (i) N ∗ is transitive and of the same type as N ∗ . (ii) If a1 , . . . , an ∈ Rp (n ≥ 0) and b ⊂ M is N ∗ -definable in a1 , . . . , an , then ap1 , . . . , apn ∈ N ∗ (where api = F −1 (ai )) and bp is N ∗ -definable in ap1 , . . . , apn by the same definition. (iii) Each x ∈ N ∗ is N ∗ -definable from parameters in Mp ∪ Dp . Moreover, if A is a solid model of L(p), then πpA ∪ F p extends uniquely to a π ⊃ πpA ∪ F p s.t. π : N ∗ ≺ N ∗ . Proof . We use the following: Fact For any X ⊂ M the following are equivalent: (a) X ≺ hM, ai for all a ∈ Rp . (b) Let Y = the smallest Y ≺ N ∗ s.t. X ∪ Rp ⊂ Y . Then Y ∩ M = X. ((b) → (a) is trivial. (a) → (b) follows from the fact that each z ∈ Y is N ∗ -definable from parameters in X ∪ Rp .) Let Ỹ = the smallest Ỹ ≺ N ∗ s.t. M ∪ Rp ⊂ Ỹ . Then Ỹ has cardinality β in V. Hence, if – in some extension V[G] – A is a solid model of L(p), then Ñ ∗ ∈ N ⊂ A, jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in L-Forcing jensen 75 ∼ where π̃ : Ñ ∗ ↔ Ỹ is the transitivation of Ỹ . Working in A, we now form Z = the smallest Z ≺ Ñ ∗ s.t. X ∪ Rp ⊂ Z, where X = rng(πpA ). Transitivize Z to get ∼ π : N ∗ ↔ Z. Then Z ∈ HωA1 = HωV1 omit: where X = rng(πpA ). Claim 1 N ∗ satisfies (i)–(iii). ∼ Proof . Let π = π̃π : N ∗ ↔ Y = the smallest Y ≺ N ∗ s.t. X ∪ Rp ⊂ Y . Then π ↾ Mp = πpA , since X = Y ∩ M by the above Fact. For a ∈ Rp we have π −1 (a) = π −1 ′′ (X ∩ a) = (πpA )−1 ′′ (X ∩ a) = ap . Thus π ⊃ πpA ∪ F p . Using this, (i)–(iii) follow easily. Claim 2 At most one N ∗ satisfies (i)–(iii). Proof . Let N ∗0 , N ∗1 be two different ones. Then (1) Let x1 , . . . , xn ∈ Mp , b1 , . . . , bm ∈ Dp . Then N ∗0  ϕ(~x, ~b ) ↔ N ∗1  ϕ(~x, ~b ). Proof . Let bi = api , ai ∈ Rp . Set: c = {h~x i ∈ M | N ∗  ϕ(~x, ~a )}. Then by (ii): cp = {h~x i ∈ Mp | N ∗h  ϕ(~x, ~b )}. QED(1) ∼ p But it then follows straightforwardly that id ↾ (Mp ∪ D ) extends to a σ : N ∗0 ↔ N ∗1 . Hence σ = id, since the models are transitive. QED(Claim 2) In the proof of Claim 1, we have shown that, if A is a solid model of L(p), then πpA ∪ F p extends to a π : N ∗ ≺ N ∗ . It remains only to note that π is unique, since every z ∈ rng(π) is N ∗ -definable from elements of X ∪ Rp = rng(πpA ∪ F p ). QED(Lemma 4) Note Clearly Mp = M , where N ∗ = hH, M , <, . . .i. We now define: Definition P = PL is revisable iff for sufficiently large cardinals Ω > 2β : Let N ∗ = hHΩ , M, <, P, . . .i where < well orders HΩ . Let p conform to N ∗ and set N ∗ = N ∗ (p, N ∗ ). Let G be P-generic over N ∗ , where N ∗ = hH, M , <, P, . . .i. Then there is q ∈ P s.t. Mq = Mp , Cq = C G , and F q = F p . (In other words q = hhMp , C G i, F p i ∈ P.) Lemma 5.1 Let P be revisable. Then P adds no new reals. ◦ f : ω → 2. W Claim ∆ = {p | f p Proof . Let ◦ f = fˇ} is dense in P. Let r ∈ P. Pick Ω big enough to verify revisability and set N ∗ = hHΩ , M, < ◦ , P, f , r, . . .i. Let p conforms to N ∗ . Set N ∗ = N ∗ (p, N ∗ ). Let N ∗ = hH, M , < , P, f , r, . . .i. Let G ∋ r be P-generic over N ∗ . Let f = f G . Let q = hhM , C G i, F p i ∈ P. ◦ Claim q ≤ r and q f = fˇ. Proof . Let A be a solid model of L(q). Let σ ⊃ πqA ∪ F q s.t. σ : N ∗ ≺ N ∗ . July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 76 jensen Subcomplete Forcing and L-Forcing (1) q ≤ r. Proof . Let C = C G , r0 = r0 ⊳ hM , Ci = q0 and πr0 ,q0 = πrG . But Rr ⊂ Rq , since r ∗ is N -definable. Let ha, ai ∈ F r , ha, a′ i ∈ F q . Claim πr0 ,q0 : hMs , ai ≺ hMq , a′ i. This is clear, since a′ = (F q )−1 (a) = σ −1 (a) and hence ha, a′ i ∈ σ −1 ′′ F r = F r ). QED(1) (2) Let s ∈ G, s = σ(s). Then A |= L(s). Proof . s0 = r0 ⊳ q0 = hM , Ci ⊳ hM, Ċ A i, and πsA0 = σ ◦ πs0 ,q0 . Let ha, ai ∈ F s . Then a = σ(a′ ), where ha′ , ai ∈ F s . Hence, πsA0 : hMs , ai ≺ hM, ai. QED(2) ◦ f = fˇ. (3) q Suppose not. Then there is i s.t. f (i) = h and q 6 q′ ◦ ◦ f (ǐ) = ȟ. Let q ′ ≤ q s.t. f (ǐ) 6= ȟ. Let A be a solid model of L(q ′ ), hence of L(q). Let s ∈ G s.t. ◦ s P f (ǐ) = ȟ. Let σ be as above. Let s = σ(s). Then q ≤ s and s f (ǐ) = ȟ. Hence q ′ , s are incompatible. But A  L(q ′ ) ∪ L(s). Contradiction! by Lemma 3.1. QED(Lemma 5.1) Now let Lc be L with its axioms reduced to the core axioms. (Thus Lc is uniquely determined by Γ, Π.) By Lemma 5.1 we have: Lemma 5.2 Let P be revisable. Let G be P-generic. Let p ∈ G. Set: A = V[G] hHκ , C G i, where κ > 2β is regular. Then A models Lc (p). An examination of the proof of Lemma 4 shows, however, the proof of the final clause in that Lemma used only that A models Lc (p). Hence: Corollary 5.3 Let P be revisable. Let G be P-generic. Let p ∈ G where p conforms to N ∗ = hHΩ , M, <, . . .i. Let N ∗ = N ∗ (p, N ∗ ). There is a unique σ ⊃ πpG ∪ F p s.t. σ : N ∗ ≺ N ∗. Proof . πpG = πpA where A is as in Lemma 5.2. QED(Corollary 5.3) Combining this with the proof of Lemma 5.1 we get: Lemma 5.4 Let P be revisable. Let N ∗ = hHΩ , M, <, P, . . .i where Ω verifies revisability. Let p conform to N ∗ . Set: N ∗ = N ∗ (p, N ∗ ) = hH, M , <, P, . . .i. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in L-Forcing jensen 77 Let G be P-generic over N ∗ and set: q = hhMp , C G i, F p i. Let G ∋ q be P-generic. Let σ ⊃ πpG ∪ F p s.t. σ : N ∗ ≺ N ∗ . Then σ ′′ G ⊂ G. (Hence σ extends uniquely to σ ∗ : N ∗ [G] ≺ N ∗ [G] with σ ∗ (G) = G.) Proof . The proof of (2) in Lemma 5.1 made use of a solid model A of L(q). An examination of this proof shows, however, that it is enough that A models Lc (q). V[G] Hence we can take A = A, where A = hHκ , C G i is as above. Hence πqA = πqG ∗ and if σ ⊃ πqG ∪ F q is s.t. if σ : N ≺ N ∗ then A |= L(s) whenever s ∈ G and s = σ(s). If s 6∈ G, there would be p ∈ G incompatible with s. But A |= L(p)∪L(s). Contradiction! QED(Lemma 5.4) We say that L is modest if all of its axioms can be forced by PL -more precisely: Definition Let L satisfy the core axioms. L is modest iff whenever G is PL -generic V[G] there is a regular κ > 2β s.t. A = hHκ , C G i satisfies L. Lemma 5.2 says that Lc is modest. Assuming modesty, we have a simple criterion for deciding whether a given condition lies in a generic set G: Lemma 5.5 Let P = PL where L is modest. Let G be P-generic. Let p ∈ P. Then p ∈ G iff the following hold: • p0 ⊳ hM, C G i, • πpG0 : hMp , ai ≺ hM, ai whenever ha, ai ∈ F p . Proof . (→) is trivial. We prove (←). Let κ be regular s.t. κ > 2β and A = hHκ , C G i satisfies L. Then A  L(p). Iff p ∈ / G there would be a q ∈ P s.t. p, q are incompatible. But A  L(p) ∪ L(q). QED(Lemma 5.5) Note In [LF], §4 we have shown that the assumption of modesty can be omitted from Lemma 5.5 assuming that P adds no reals. This is because P = PL∗ , where V[G] L∗ is the set of L statements forced to hold in A = hHκ , C G i, where κ > 2β is regular. We shall not use that here, however, since our languages will always be modest. (We are unlikely to adopt an axiom without the expectation that it will be forced.) Finally, we note that there is an apparently weaker notion of revisability relative to a parameter: Definition P is weakly revisable iff there exist a cardinal Ω > 2β and an s ∈ HΩ s.t. whenever N ∗ = hHΩ , M, <, P, s, . . .i and p conforms to N ∗ , then, letting N ∗ = N ∗ (p, N ∗ ) = hH, M , <, P, s, . . .i, we have: Let G be P-generic over N ∗ . Then q = hhM , C G i, F p i ∈ P. It turns out that this is equivalent to full revisability. This fact is useful (and may be used tacitly) in verifying revisability. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 78 jensen Subcomplete Forcing and L-Forcing Lemma 5.6 Let P be weakly revisable. Then it is fully revisable. Proof . Let Ω be the smallest cardinal verifying weak revisability. Let Ω′ > H Ω be a cardinal. Let N ′∗ = hHΩ′ , M, <′ , P, . . .i. Let p conform to N ′∗ and let N ′∗ = N ∗ (p, N ′∗ ) = hH ′ , M ′ , <, P′ , . . .i. Let G be P′ -generic over N ′∗ . Claim q = hhM ′ , C G i, F p i ∈ P. Note that Ω, s are N ′∗ -definable, where s = the <′ -least s s.t. hΩ, si verifies weak revisability. Let A be a solid model of L(p) and let σ ′ ⊃ πpA ∪F p s.t. σ ′ : N ′∗ ≺ N ′∗ . Let σ ′ (Ω, s) = Ω, s. (1) Set: N ∗ = hHΩ , M, <, P, s, . . .i where <=<′ ∩HΩ2 . Then p conforms to N ∗ . Set: N ∗ = N ∗ (p, N ∗ ) = hH, M , <, P, s, . . .i. Let σ ⊃ πpA ∪ F p s.t. σ : N ∗ ≺ N ∗ . Then each x ∈ rng(σ) is N ∗ -definable in parameters from rng(πpA ∪ F p ). Hence it is N ′∗ -definable in these parameters. Hence: rng(σ) ⊂ rng(σ ′ ). (2) But: (3) M = Mp = M ′ ; σ ↾ M = πpA = σ ′ ↾ M . Moreover, each a ∈ P(M ) ∩ N ∗ is hM , bi-definable from parameters from M , where b ∈ Dp . Similarly for P(M ) ∩ N ∗ . Hence: (4) P(M ) ∩ N ∗ = P(M ) ∩ N ′∗ . Since σ ↾ Dp = F p = σ ′ ↾ Dp and σ ↾ M = σ ′ ↾ M ′ , we conclude (5) σ ↾ P(M ) = σ ′ ↾ P(M ). P = h|P|, ≤P i is canonically codable as a subset of P(M ). Similarly for P′ . But σ(P) = σ ′ (P′ ) = P. It follows easily that. (6) P = P′ and σ ↾ P = σ ′ ↾ P′ . But if ∆ ∈ P(P) ∩ N ∗ , then ∆ = (σ −1 ) · σ(∆) ∈ N ′∗ . Hence: (7) P(P) ∩ N ∗ ⊂ N ′∗ . Hence G is generic over N ∗ and we conclude: (8) q = hhM , C G i, F p i ∈ P. QED(Lemma 5.6) In conclusion we say a few words about the difference between the present apA proach and that taken in [LF]. There too we approximated M = LA τ s.t. Lτ = Hβ for some β > ω1 . Our intention, however, was simply to make M the limit of a tower of countable models. In place of an approximation system Γ, Π we worked with a collection T of tower segments hhMi | i ≤ αi, hπij | i ≤ j ≤ αii satisfying: Mi Mi i • Mi = LA βi , i ≤ ω1 , Mh ∈ Hω1 for h < i. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 79 L-Forcing • πij : Mi ≺ Mj (i ≤ j) with πii = id. • πij πhi = πhj . S rng(πiλ ). • If λ ≤ α is a limit ordinal, then Mλ = i<λ We sometimes imposed further requirements on T , but T was always primitive recursive. For t ∈ T we set: t t = hhMir | i ≤ αt i, hπij | i ≤ j ≤ αt ii. Call t a segment of s iff αt ≤ αs and Mit = Mis , t s πij = πij for i ≤ j ≤ αt . ◦ Our language contained a single constant t in addition to x (x ∈ N ) and the core axioms:  _ _ ◦ V  ZFC− , Hω1 = H ω1 , w = z , t ∈ T, v v∈x↔ α ◦ t = ω1, Mωt 1 = M, V i< z∈x t ω 1 M i ∈ Hω 1 . We now show how to convert this approach into our present one. For each t ∈ T set: et = hMαt , {hy, x, ii | i < αt ∧ πiαt (x) = y}i. Set Γ = {et | t ∈ T }. Note that t is uniquely recoverable from et . We set: et ⊳ es iff et is a segment of e1 , π : et ⊳ es iff (et ⊳ es and π = παs t ,αs ). Then Γ, Π is an approximation system and the above core axioms translate into our usual core axioms. July 21, 2012 15:2 80 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen Chapter 6 Examples We now display some specific examples of L-forcing. All of them are revisable and will turn out to be subcomplete as well. 6.1 Example 1 Assume CH and 2ω1 = ω2 . Without adding reals we wish to make ω2 become ω-cofinal. We first define our approximation system: Definition Γ = the set of hM, Ci s.t. − • M = LA τ models ZFC and “ω1 is the largest cardinal”. • C is a cofinal subset of OnM of order type ω. Definition For u ∈ Γ set u = hMu , Cu i. Definition Π = the set of hπ, u, vi s.t. u, v ∈ Γ, π : Mu ≺ Mv , π ′′ Cu = Cv . W We again write π : u ⊳ v for hπ, u, vi ∈ Π and u ⊳ v for π π : u ⊳ v. Definition αu = ω1Mu for u ∈ Γ. We note that: (1) Let v ∈ Γ. Let α ≤ αv . There is at most one u ∈ Γ s.t. u ⊳ v and α = αu . Thus ⊳ is a tree. Now let M = LA ω2 , where Lω2 [A] = Hω2 . Set: N = hHω3 , M, <, . . .i where < well orders Hω3 . Let L be the language on N constaining exactly the core axioms (wrt. Γ, Π). Lemma 1 L is consistent. Proof . Let θ > 2β be a regular cardinal. Let H = Hθ and σ : H ≺ H, where H S is countable and transitive. Let σ(M , N ) = M, N , σ(L̃) = L. Set M̃ = σ(u). u∈M Then σ ↾ M : M ≺ M̃ cofinally. Let hH̃, σ̃i be the liftup of hM , σ ↾ M i. Then 81 July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 82 Subcomplete Forcing and L-Forcing σ̃ : H ≺ H̃ ω2M -cofinally. Let k̃ : H̃ ≺ H s.t. kσ̃ = σ, k ↾ ω2H̃ = id. Let k(L̃) = L. Then L̃ is a language on Ñ and it suffices to show: Claim L̃ is consistent. Proof . Let C ⊂ M be cofinal in OnM with order type ω. Set C̃ = σ ′′ C. Then QED(Lemma 1) hHωH̃2 , C̃i models L̃. Now let P = PL . We show that P satisfies a particularly strong form of revisability. Lemma 2 Let p ∈ P. Let C be cofinal in OnMp with order type ω. Then q = hhMp , Ci, F p i ∈ P. Proof . Let A be a solid model of L(p). We shall “resection” A to get a solid model A′ of L(q). Let A = h|A|, C A i. Set A′ = h|A|, C ′ i where C ′ = πpA ′′ C. Since C ′ is defined in A, we have A′  (ZFC− ∧ H ω1 = Hω1 ). Since HωA1 = Hω1 it follows easily that whenever X ⊂ M is countable in A, then there is u ⊳ hM, C ′ i s.t. u ∈ Hω1 and X ⊂ rng(πuA ). Hence all core axioms hold. QED(Lemma 2) An immediate corollary is: Corollary 2.1 P is revisable. Thus, if G is P-generic and κ > 2β is regular, hHκG , C c i satisfies all core axioms. But these are exactly the axioms of L. Hence L is modest. Making use of Lemma 2 we now prove: Lemma 3 P is subcomplete. ω2 − Proof . Let θ > 22 . Let W = LA τ be a ZFC model s.t. Hθ ⊂ W and θ < τ . Let π : W ≺ W , where W is countable and full. Let π(θ, P, s) = θ, P, s. Since ω2 ≤ δ(P) it suffices to show: Claim Let G be P-generic over W . There is q ∈ P s.t. whenever G ∋ q is P-generic, then there is σ ∈ V[G] s.t. (a) (b) (c) (d) σ:W ≺W σ(P, θ, s) = P, θ, s CωW2 (rng(σ)) = CωW2 (rng(π)) σ ′′ G ⊂ G. ∼ Now let C = CωW2 (rng(π)), k : W̃ ↔ C, where W̃ is transitive. Set π̃ = k −1 · π. Then π̃ : W ≺ W̃ is ω3W -cofinal. If σ satisfies (a)–(d) and we set: σ̃ = k −1 σ, then σ̃ : W ≺ W̃ is also ω3W -cofinal. But since σ̃ takes ω2W cofinally to ω2 = ω2W̃ , it follows that σ̃ is ω2 -cofinal. The following lemma hints at the possibility of such a σ̃: Let π̃(θ, P, s) = θ̃, P̃, s̃. Sublemma 3.1 Let δ = δW̃ = the least δ s.t. Lδ (W̃ ) is admissible. Then the following language L̃ on Lδ (W̃ ) is consistent: jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 83 Examples Predicate: ∈ ◦ Constants: σ , x (x ∈ Lδ (W̃ )) V W W ◦ Axioms: ZFC− , v(v ∈ x ↔ v = z), σ : W ≺ W̃ ω W̃ 2 -cofinally, z∈x σ(P, θ, s) = P̃, θ̃, s̃. Proof . Let hŴ , π̂i be the liftup of hW , π ↾ Hi, where H = (Hω2 )W . (Hence π ↾ H = π̃ ↾ H.) Let k̂ : Ŵ ≺ Ñ s.t. k̂π̂ = π̃, k̂ ↾ ω2Ŵ = id. Then k̂ is cofinal in W̃ . Let δ̂ = δŴ be least s.t. Lδ̂ (Ŵ ) is admissible. Let L̂ be defined on Lδ̂ (Ŵ ) as L̃ was defined on Lδ (W̃ ) with Ŵ , P̂, θ̂, ŝ in place of W̃ , P̃, θ̃, s̃, where π̂(P, θ, s) = P̂, θ̂, ŝ. It suffices to show: Claim L̂ is consistent. This is trivial, however, since hŴ , π̂i models L̂. β Now let Ω > 2 be a cardinal. Set: N conform to N ∗ and set: ∗ QED(Sublemma 3.1) = hHΩ , M, W, P, θ, s, π, . . .i. Let p N ∗ = N ∗ (p, N ∗ ) = hH ′ , M ′ , W ′ , P′ , θ′ , s′ , π ′ , . . .i. Let W̃ ′ , π̃ ′ , L̃′ be defined in N ∗ as W̃ , π̃, L̃ were defined in N ∗ . Since N ∗ is ◦ countable, there is a solid model A of L̃. Set σ̃ = σ A . Then σ̃ : W ≺ W̃ ′ ω2W -cofinally. ′ Hence W̃ ′ = CωW̃W ′ (rng(σ̃)). Set: C = C G , C ′ = π ′ ′′ C. Then C ′ is cofinal in 2 ′ ω2W and has order type ω. Set: q = hhM ′ , C ′ i, F p i. Then q ∈ P by the strong revisability lemma. Let G ∋ q be P-generic. Let π ∗ ⊃ πqG ∪ F q s.t. π ∗ : N ∗ ≺ N ∗ . Let π ∗ (k ′ ) = k. Set: σ ′ = k ′ σ̃, σ = π ∗ σ ′ . Then σ ∈ V[G]. Claim σ satisfies (a)–(d). ′ Proof . (a), (b) are trivial. We prove (c). Set ω2′ = ω2W . ′ ′ CωW′ (rng(σ ′ )) = CωW′ (rng(π ′ )), (1) 2 2 ′ ′ since k ′ ′′ W̃ ′ = CωW′ (rng(π ′ )) by definition and k ′ ′′ W̃ ′ = CωW′ (rng(σ ′ )), since W̃ ′ = 2 2 ′ CωW̃′ (rng(σ̃)), k ′ ′′ rng(σ̃) = rng(σ ′ ), and k ′ ↾ ω2′ = id. 2 (2) CωW2 (rng(σ)) ⊂ CωW2 (rng(π)), ′ ′ since rng(σ) = π ∗ ′′ rng(σ ′ ) ⊂ π ∗ ′′ CωW′ (rng(π ′ )) ⊂ π ∗ (CωW′ (rng(π ′ )) = CωW2 (rng(π)). 2 (3) 2 CωW2 (rng(π)) ⊂ CωW2 (rng(σ)), ′ since rng(π) = π ∗ ′′ rng(π ′ ) ⊂ π ∗ ′′ CωW2 (rng(σ ′ )) ⊂ CωW2 (rng(σ)), since π ∗ ′′ rng(σ ′ ) = rng(σ) and π ∗ ′′ ω2′ ⊂ ω2 . QED(c) We now prove (d). Since L is modest we have: Sublemma 3.2 Let C = C G . Then G = GC = the set of p ∈ P s.t. p0 ⊳ hM, Ci and π : hMp , ai ≺ hM, ai whenever ha, ai ∈ F p , when π = πp0 ,hM,ai . July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 84 jensen Subcomplete Forcing and L-Forcing Now let r ∈ G, r = σ0 (r). Then r0 = r 0 ⊳ hM , Ci and πr0 ,hM,Ci = πrG , where C = C G . Obviously, σ ′ ↾ M : hM , Ci ⊳ hM ′ , C ′ i and πqG : hM ′ , C ′ i ⊳ hM, Ci, where πqG = π ∗ ↾ M ′ . Hence σ ↾ M : hM , Ci ⊳ hM, Ci. Let r = σ(r). Then r0 = r 0 and F r = {hπ ∗ (a), ai | ha, ai ∈ F r }. Clearly r0 ⊳hM, Ci and: πr0 ,hM,Ci = σ ◦ πrG . Now let ha, ai = hπ ∗ (a′ ), ai ∈ F r . Then πrG : hMr , ai ≺ hM ′ , a′ i and σ(hM ′ , a′ i) = hM, ai. QED(Lemma 3) Note We could in this case have omitted the predicate C and simply taken Γ as − the set of hM, ∅i s.t. M = LA and “ω1 is the largest cardinal”. Π τ models ZFC would then be defined as the set of hπ, u, vi s.t. u, v ∈ Γ, π ′′ Mu ≺ Mv cofinally. If we call P′ the resulting set of conditions, then it is the “same” as P in the sense that BA(P) ≃ BA(P′ ). Note P is, in fact, equivalent to Namba forcing in the sense that BA(P) ≃ BA(N). This is surprising, since P not only looks different and has a different motivation, but the combinatorics involved in the proofs are quite different. 6.2 Example 2 β Now let β > ω2 be a cardinal and assume: 2ω = ω1 , 2ω1 = ω2 , 2 ` = β. We shall develop a forcing very much like the previous forcing which, however, gives cofinality ω not only to ω2 but to every regular τ ∈ [ω2 , β]. There will be some variation in the definition of the forcing, depending on whether cf(β) = ω1 . Thus, in this example, we assume cf(β) = ω1 . In Example 3 we shall then detail the changes which must be made if cf(β) 6= ω1 . Let M = LA β where Hω2 = Lω2 [A] and Hβ = Lβ [A]. M is then smooth in the sense defined in Chapter 3.2. ⋆⋆⋆⋆⋆ Definition Relable the classes Γ, ⊳ defined in Example 1 as Γ0 , ⊳0 . Set: Γ = the collection of hM, Ci s.t. • M = LA β is smooth. • γ = ω2M exists and Lγ [A] = Hω2 in M . • C ⊂ γ, sup C = γ, otp C = ω. For u = hMu , Cu i ∈ Γ set: αu = αMu = ω1Mu , γu = γMu = ω2Mu , u Mu = LA βu , u Mu0 = LA γu , u0 = hMu0 , Cu i, July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Examples jensen 85 Hence: u0 ∈ Γ0 . Definition Let u, v ∈ Γ. π : u ⊳ v iff the following hold: • π 0 : u0 ⊳0 v 0 where π 0 = π ↾ Mu0 . • π : Mu ≺ Mv . • Let π : Mu →Σ0 Muv cofinally. Then hMuv , πi is the liftup of hMu , π 0 i. (In other words π : Mu →Σ0 Muv γu -cofinally.) hΓ, Πi is easily seen to be an approximation system. ⋆⋆⋆⋆⋆ We return to M = LA β as stated at the outset. Let N = hHβ + , M, <, . . .i where < well orders Hβ + . Let L be the language on N containing only the core axioms (wrt. Γ, Π). Lemma 4 L is consistent. Proof . Let θ > 2β be a regular cardinal. Let π : H ≺ Hθ s.t. H is countable and transitive and π(M , N , L) = M, N, L. Let Ĥ = HωM2 and set hH̃, π̃i = the liftup of hH, π ↾ Ĥi. Let k : H̃ ≺ Hθ s.t. kσ̃ = σ. k ↾ ω2H̃ = id. Set M̃ , Ñ , L̃ = π̃(M , N , L). Then k(L̃) = L and it suffices to show: Claim L̃ is consistent. Let C ⊂ ω0M cofinally s.t. otp(C) = ω. Set C̃ = π ′′ C = π̃ ′′ C. We prove: Claim hHω2 , C̃i models L̃. Proof . All axioms are trivial except for the last one. We show that if X ⊂ M is countable, then there is u ∈ Γ ∩ Hω1 s.t. u ⊳ hM, C̃i and X ⊂ rng(πu,hM,C̃i ). We ∼ construct such a u: Let Z ≺ H̃ be countable s.t. X ∪ rng(π̃) ⊂ Z. Let π ′ : H ′ ↔ Z. ′ ′−1 ′ ′−1 ′′ ′′ ′ ′ ′ Set: M = π (M̃ ), C = π C̃, π = π ↾ M . Then X ⊂ rng(π ) and it suffices to show: Claim π ′′ : hM ′ , C ′ i ⊳ hM̃ , C̃i. π ′ ↾ M 0 : hM 0 , C ′ i ⊳0 hM̃ 0 , C̃i is obvious. We therefore need only to show: Claim Let π ′′ : M ′ →Σ0 M ∗ cofinally. Then the map π ′′ is ω2 M ′ -cofinal into M ∗ . ′ Proof . First note that π ′ : H ′ ≺ H̃ ω2H -cofinally, since if x ∈ H̃, then x ∈ π̃(u), where u < ω2 in H. Set u′ = π ′−1 π̃(u). Then x ∈ π ′ (u′ ), u′ < ω2 in H ′ . Now let x ∈ M ∗ . By cofinality there is v ∈ M ′ s.t. x ∈ π ′′ (v). Let u ∈ H̃ s.t. x ∈ π ′ (u) and u < ω2 in H ′ . Set: w = u ∩ v. Then x ∈ π ′′ (w) where w < ω in M ′ , since M ′ = Hβ ′ in H ′ , where β ′ = (π ′ )−1 (β̃). QED(Lemma 4) We then define P = PL as before. Exactly as before we get: Lemma 5 Let p ∈ P. Let C ⊂ γp be cofinal in γp with order type ω, where M γp =Df γp0 = ω2 p . Then q = hhMp , Ci, F p i ∈ P. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 86 jensen Subcomplete Forcing and L-Forcing Hence: Corollary 5.1 P is revisable. Hence hHκ [G], C G i models the core axioms whenever G is P-generic and κ > 2β . But L has only the core axioms and is, therefore, modest. Using this we obtain: Lemma 6 P is subcomplete. The proof is virtually identical to that of Lemma 3. However, in the verification of (d) at the end of the proof we need additional justification for: σ ′ ↾ M : hM , Ci ⊳ hM ′ , C ′ i. Letting σ ′ ↾ M map M cofinally to M ∗ , we must show: σ ′ ↾ M : M −→Σ0 M ∗ ω2M -cofinally. This follows from σ ′ : N ≺ Ñ ′ ω2N -cofinally by the argument used in Lemma 4 to get: π ′′ : M ′ −→Σ0 M ∗ ′ ω2M -cofinally ′ from: π ′ : H ′ ≺ H̃ ′ ω2H -cofinally. QED(Lemma 6) P obviously collapses β to ω1 . We now show that its successor is not collapsed: Lemma 7 Let G be P-generic. Then β + is regular in V[G]. This is immediate from: Sublemma 7.1 B = BA(P) has a dense subset of size β. Proof . We defined a collection S of statements in the forcing language s.t. S ≤ β (in V), and for each p ∈ P there is a ψ ∈ S s.t. 0 6= [[ψ]] ⊂ [p]. ([p] being the ◦ ◦ smallest a ∈ B s.t. p ∈ a.) Let C be the canonical term s.t. C G = C G for P-generic G. For each triple hu, a, ai s.t. u = hMu , Cu i ∈ Γ ∩ Hω1 , a : ω → P(Mu ), a : ω → M, let ψuaa be the statement: ◦ ǔ ⊳ hM̌ , Ci ∧ ◦ where π = π ◦ ǔhM̌, C i ^ i<ω ^ ◦ z(z ∈ ǎ(i) ←→ π (z) ∈ ǎ(i)) . All such triples are elements of M , so the set S of such statements has at most cardinality β. We now show that for each p ∈ P there is ψ ∈ S with 0 6= [[ψ]] ⊂ [p]. It suffices to prove this for a dense subset of P, so assume w.l.o.g. that p conforms to N ∗ = hH(2β )+ , M, <i. Let G ∋ p be P-generic. p Let β̃ = sup πpG ′′ βp . Then β̃ < β. Set M̃ = LA , where M = LA β . For each a ∈ R β̃ set ã = a ∩ M̃ . Let hhai , ai i | i < ωi enumerate F p in V. Set a = hai | i < ωi, July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 87 Examples ã = hãi | i < ωi. Let ψ = ψp0 ,a,ã . Then [[ψ]] 6= 0, since ψ is true in V[G]. We claim that [[ψ]] ⊂ [p], or equivalently: Claim Let G be P-generic. Then G ∩ [[ψ]] 6= ∅ → p ∈ G. Then p0 ⊳ hM, Ci, since ψ is true in V[G]. Let ha, ai ∈ F p . We must show: Claim π : hMp , ai ≺ hM, ai, where π = πp0 ,hM,Ci . Set: b = {hz1 , . . . , zn i | hM, ai  X (z1 , . . . , zn )}. Then b ∈ Rp , since p conforms to N ∗ . Moreover hb, bi ∈ F p where b has the same definition over hMp , ai. Hence: hMp , ai  X (z1 , . . . , zn ) ←→ hz1 , . . . , zn i ∈ b ←→ π(hz1 , . . . , zn i) ∈ b̃ = M̃ ∩ b −→ π(hz1 , . . . , zn i) ∈ b −→ hM, ai  X (π(z1 ), . . . , π(zn )). Since this holds for all X we have: π : hMp , ai ≺ hM, ai. 6.3 QED(Lemma 7) Example 3 β We now assume 2ω = ω1 , 2ω1 = ω2 , 2 ` = β, and cf(β) 6= ω1 . We again want to give cofinality ω to all regular cardinals τ ∈ [ω2 , β]. It is clear that β will also acquire cofinality ω, since it either already has cofinality ω, or its cofinality lies in [ω2 , β). The simplest way of handling this is to revise the definition of ⊳ to: Definition Let u, v ∈ Γ. π : u ⊳ v iff the following hold: • π 0 : u0 ⊳0 v 0 where π 0 = π ↾ Mu0 . • π : Mu ≺ Mv γu -cofinally. Let M = LA β where Lβ [A] = Hβ . As before set N = hHβ + , <, M, . . .i. Let L be the language on N with only the core axioms. Exactly as before we prove: Lemma 8 L is consistent. (Note that if N is countable and transitive, π : N ≺ N , π(M ) = M , and M 0 = HωM2 , then if hÑ , π̃i is the liftup of hN , π ↾ M 0 i, then π ↾ M : M ≺ M̃ ω2M -cofinally, where M̃ = π̃(M ).) We then set P = PL . Exactly as before we get: Lemma 9 P is strongly revisable. Corollary 9.1 P is revisable. Hence L is modest, since it has only the core axioms. Exactly as before we get: Lemma 10 P is subcomplete. Lemma 7 does not go through, however. In fact 2β acquires cardinality ω1 . This follows from the very general theorem: July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 88 jensen Subcomplete Forcing and L-Forcing Lemma 11 Let W be an inner model of ZFC and CH. Let Hω1 = HωW1 . Let β > ω1 s.t. 2 in V. β ` = β in W . Suppose that cf(β) = ω and β = ω1 in V. Then card(2β ) = ω1 Proof . Let M = LA β where Lβ [A] = Hβ in W . Let f map ω1 onto M in V. Let hβi | i < ωi ∈ V be cofinal in β. Set: Xα = f ′′ α for α < ω1 . Set: C = {α < ω1 | α = ω1 ∩ Xα ∧ Xα ≺ M ∧ {βi | i < ω} ⊂ Xα }. ∼ For α ∈ C set πα : Mα ↔ Xα , where Mα is transitive. Then Mα ∈ Hω1 . For any B ⊂ β, there is α ∈ C, s.t. B ∩ βi ∈ Xα for i < ω. Set: [ B = {πα−1 (B ∩ βi ) | i < ω}. Then hα, Bi ∈ Hω1 and B is recoverable from hα, Bi by: [ πα (u ∩ B). π̃(α, B) = u∈Mα Thus π̃ maps a subset of Hω1 onto P(β). 6.4 QED(Lemma 11) The extended Namba problem Shelah was the first to show that Namba forcing can be iterated without adding reals. If we iterate it out to a strongly inaccessible κ, then κ becomes the new ω2 and arbitrarily large regular cardinals below κ become ω-cofinal. However, many regular cardinals become ω1 -cofinal. The “extended Namba problem” asks whether, without adding reals, one can make κ become ω2 while giving all of the regular cardinals in the interval (ω1 , κ) cofinality ω. This problem seemed so difficult that at one point we conjectured a provably negative answer in ZFC for all κ. Moti Gitik then disproved this conjecture by constructing a ZFC model in which the extended Namba problem had a positive solution for some κ. His model was a generic extension of a universe containing a supercompact cardinal. Following Gitik’s breakthrough we then obtained a positive solution in ZFC for all κ. It is impossible to give the full proof of that result in these notes, but we shall endeaver to give some account of the methods used. We may assume w.l.o.g. that GCH holds below κ, since we may achieve this by a prior forcing in which all collapsed regular cardinals acquire a cofinality ≥ ω2 . If we then give the surviving regular cardinals in (ω1 , κ) the cofinality ω, the collapsed ones will also become ω-cofinal. It is natural to try to solve this problem by an iteration hBi | i ≤ κi. We ask now what the initial steps of this iteration should look like. We follow the convention that B0 = 2. Thus B1 is the first stage which “does something”. We certainly expect it to give ω2 the cofinality ω without adding reals. By Lemma 11 it follows that ω3 will be collapsed, so ω3 must acquire cofinality ω. But then ω4 is collapsed etc. Thus every ωn must be collapsed with cofinality ω. By Lemma 11, it then July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Examples jensen 89 follows that ωω+1 is collapsed etc. This chain of implications does not break down until we reach ωω1 . There, however, it does break down, since we can use the P of Example 2 with β = ωω1 . All regular cardinals in (ω1 , ωω1 ) acquire cofinality ω and ωω1 +1 is not collapsed, thus becoming the new ω2 . We take B1 ≃ BA(P). We can then repeat the process, getting B2 ⊇ B1 which collapses ωω1 ·2 to ω1 etc. This gives us the first ω stages hBi | i < ωi. Our job now is to find an appropriate limit Bω . Since each Bi is subcomplete, the inverse limit B∗ is also subcomplete. However, a bit of reflection shows that B∗ is too small to do the job: At the limit stage ωω1 ·ω will be collapsed to ω1 . Hence by Lemma 11 ω(ω1 ·ω)+1 will be collapsed and hence must acquire cofinality ω etc. Proceeding in this fashion we see that ωω1 ·(ω+1) must be collapsed. Thus our limit algebra must be large, not containing any dense set of size less than ωω1 (ω+1) . At the same time it should have a dense subset of size ωω1 (ω+1) in order that the successor is preserved. It turns out that a limit with the requisite properties can be obtained by a construction rather like that of Example 2. We shall now sketch that construction, but a full verification of its properties is beyond the purview of these notes. Let M 0 = LA γ where γ = ωω1 ω , Lγ [A] = Hγ , and A canonically codes hBi | i < ωi. We define Γ0 , Π0 as follows: Γ0 = the set of u = hMu , Bu i s.t. u • M = LA γu where Mu models Zermelo set theory and Au canonically codes a sequence hBui | i < ωi of complete Boolean algebras in the sense of M with Bui ⊆ Buj (i ≤ j < ω). S • Bu ⊂ Bui s.t. Bu ∩ Bi is Bi -generic over M for i < ω. i • sup{δ(Bi ) | i < ω} = β and Bi collapses δ(Bi ) to ω1M for i < ω. Π0 = the set of hπ, u, vi s.t. u, v ∈ Γ0 , π : Mu ≺ Mv and π ′′ Bu ⊂ Bv . We write π : u ⊳0 v for hπ, u, vi ∈ Π0 . Setting Bui = Bu ∩ Bui , we see that π has a unique extension π i s.t. π i : Mu [Bui ] ≺ Mv [Bvi ] and π i (Bui ) = Bvi . Set: S S Mu∗ = Mu [Bui ] and π ∗ = π i . Then π ∗ : Mu∗ → Mv∗ cofinally. i i Au Letting fui be the canonical map of ω1 onto Lδ(B i ) , we see that π is uniquely u i i characterized by: π ◦ fu = fv for i < ω. It follows easily that π = πuv is the unique π : u ⊳0 v and that Γ0 , Π0 is an approximation system. Now let M = LA β where β = ωω1 (ω+1) , Lβ [A] = Hβ and M 0 = LA (γ = ω ). Set: ω1 ·ω γ Γ = the set of u = hMu , Bu i s.t. Mu is smooth and there is γ = γu ∈ Mu u u s.t. u0 = hMu0 , Bu i ∈ Γ0 , where Mu = LA and Mu0 = LA γ . β We then set: Π = the set of hπ, u, vi s.t. u, v ∈ Γ and: • π 0 : u0 ⊳ v 0 where π 0 = π ↾ M 0 . • π : Mu ≺ Mv . July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 90 jensen Subcomplete Forcing and L-Forcing • Let π : Mu → Mu,v cofinally. Then hMu,v , πi is the liftup of hMu , π 0 i. We again set: π :u⊳v iff hπ, u, vi ∈ Π. Thus hΓ, Πi is an approximation system which is related to hΓ0 , Π0 i exactly as in Chapter 6.2. Again, letting M = LA β be as above, and N = hHβ + , <, M, . . .i, we form the language L on N containing only the core axioms. Lemma 12 L is consistent. Proof . Let B∗ = the inverse limit of hBi | i < ωi. Then B∗ is subcomplete. Let B ∗ be B∗ -generic. We prove the consistency of L in V[B ∗ ]. Let Bi = B ∗ ∩ Bi , S B= Bi . Let H = H(2β )+ in V. Let π : H ≺ H in V[B ∗ ] s.t. H is countable i<ω and transitive. Let: π(N , M , M 0 , hBi | i < ωi) = N, M, M 0 , hBi | i < ωi. Set B i = π −1 ′′ Bi for i < ω. Since we are working in V[B ∗ ] we may assume that B i is Bi -generic over M for i < ω. Clearly π takes M 0 to M 0 cofinally. Moreover: π ↾ M 0 : hM 0 , Bi ⊳0 hM 0 , Bi. Now let hH̃, π̃i be the liftup of hH, π ↾ M 0 i. Let: π̃(M , N , L) = M̃ , Ñ , L̃, where π(L) = L. Since there is k : H̃ ≺ H with k(L̃) = L, it suffices to prove that L̃ is consistent. We claim: Claim hHκ , Bi models L̃, where κ > 2β is regular in V. Proof . The only problematical case is: Let X ⊂ M̃ be countable. There is u ∈ Γ ∩ Hω1 s.t. u ⊳ hM̃ , Bi and X ⊂ rng(πu,hM̃ ,Bi ). Let Y ≺ H̃ be countable s.t. rng(κ̃) ∪ X ⊂ Y and whenever ∆ ∈ Y is dense in Bi (i < ω), then ∆ ∩ B 6= ∅. Let: ′ ∼ π ′ : H ′ ↔ Y, π ′ (M 0 , M ′ , hB′i | i < ωi) = M 0 , M̃ , hBi | i < ωi. Set: B ′ = π ′−1 ′′ Bi , π ′′ = π ′ ↾ M ′ . Claim π ′′ : hM ′ , B ′ i ⊳ hM̃ , Bi. Clearly: π ′′ ↾ M 0′ : hM 0′ , B ′ i ⊳0 hM 0 , Bi. Since π ′′ : M ′ ≺ M̃ , it suffices to show that: If π ′′ : M ′ → M ∗ cofinally, then hM ∗ , π ′′ i is the liftup of hM ′ , π ′′ ↾ M 0′ i – i.e. ′ that π ′′ takes M ′ γ ′ -cofinally to M ∗ wheres γ ′ = (ω1 · ω)M . This follows by the usual argument. QED(Lemma 12) The strong revisability lemma reads: Lemma 13 For sufficiently large θ > 2β we have: Let N ∗ = hHθ , M, P, <, . . .i. Let S M Bi p conform to N ∗ and set: N ∗ = N ∗ (N ∗ , p) = hH, M , P, <, . . .i. Let B ⊂ i<ω M p s.t. B ∩ BM i is Bi -generic over M for i < ω. Then q = hhM , Bi, F i ∈ P. July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 91 Examples We must forego the proof of Lemma 13, since it is very long and involves properties of the algebras Bi which we have not developed here. An immediate corollary is: Corollary 13.1 P is revisable, since revisability says that the above holds when ∗ B = B G for a G which is P-generic over N . Since L has only the core axioms, it is then modest. But then we get: Lemma 14 P is subcomplete. We sketch briefly the proof of Lemma 14, which is largely the same as before. Let θ be big enough to verify the subcompleteness of Bi for i < ω. Let W = LA τ be a ZFC− model with Hθ ⊂ W and θ < τ . Let π : W ≺ W where W is countable and full. Let π(θ, P, s) = θ, P, s. Claim There is q ∈ P s.t. if G ∋ q is P-generic, there is σ ∈ V[G] with: (a) σ : W ≺ W (b) σ(θ, P, s) = θ, P, s (c) CγW (rng(σ)) = CγW (rng(π)), where γ = On ∩ M 0 = sup δ(Bi ). i<ω (d) σ ′′ G ⊂ G. (Note γ ≤ δ(P), since otherwise γ would not be collapsed.) Let Ω > θ be big enough to verify the strong revisability of P. Set: N ∗ = hHΩ , <, M, N, P, W, π, . . .i. Let p conform to N ∗ . Set: N ∗ = N ∗ (N ∗ , p) ′ B′i = BM (i < ω). Set θ′ , P′ , s′ = π ′ (θ, P, s). i On ∩ M 0 . Noting that W ′ is countable and Theorem 2 we get: S Sublemma 14.1 There are σ ′ and B ′ ⊂ i<ω over W ′ for i < ω and: = hH ′ , M ′ , N ′ , P′ , W ′ , π ′ , . . .i. Set: Set γ ′ = π ′ (γ), where π(γ) = γ = imitating the proof of Chapter 4, B′i s.t. Bi′ = B ′ ∩ B′i is B′i -generic (a) σ ′ : W ≺ W ′ (b) σ ′ (θ, P, s) = θ′ , P′ , s′ ′ ′ (c) CγW′ (rng(σ ′ )) = CγW′ (rng(π ′ )) (d) σ ′ ′′ B ⊂ B ′ , where B = B G . ◦ To get this we successively define σ i , bi ∈ B′i s.t. whenever Bi′ ∋ bi is P′ -generic over ◦ ′ W ′ and σi′ = σ i Bi , then σi′ satisfies (a)–(c) and: σi′ ′′ B i ⊂ Bi′ (where B i = B ∩ Bi ). We ensure hi (bi+1 ) = bi for i < ω. We then successively choose Bi′ ∋ bi with: Bi′ S is B′i -generic over W ′ and Bi′ ⊃ Bℓ′ for ℓ < i. We set: B ′ = Bi′ and let σ ′ be the ◦ i ′ ’limit’ of σi′ = σ i Bi (i < ω) exactly as in the proof of Chapter 4, Theorem 2. QED(Sublemma 14.1) July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in 92 jensen Subcomplete Forcing and L-Forcing By the strong revisability lemma we have: q = hhM ′ , B ′ i, F p i ∈ P. Let G ∋ q be P-generic. Then πqG ∪ F q extends uniquely to: σ ∗ : N ∗ ≺ N ∗ . Set σ = σ ∗ · σ ′ . It follows by a virtual repetition of previous proofs that σ has the desired properties. QED(Lemma 14) S Bi → B′ by: Now let B′ = BA(P). We define a map µ : i<ω ◦ µ(b) = [[b̌ ∈ B]] where ◦ BG = BG for all generic G. Then: (1) µ is injective. ◦ Proof . It suffices to show: µ(b) = 0 → b = 0. Let b 6= 0. Then L + b ∈ B is consistent by the proof that L is consistent. Hence there is p ∈ P, b ∈ Bp s.t. ◦ π p (b) = b. Hence p  b̌ ∈ B – i.e. p ∈ µ(b). µ ↾ Bi is a complete embedding.  T  hh Ť ◦ii bi ∈ B Proof . µ bi = QED(1) (2) i∈X QED(2) i∈X Hence we can take B ⊃ S ∼ Bi s.t. for some k, k : B′ ↔ B and kµ = id. B is then a i<ω limit of hBi | i < ωi which collapses ̺ = ωω1 (ω1 +1) to ω1 while making all regular τ ∈ (ω1 , ̺) become ω-cofinal. A proof like that of Lemma 7 shows that ̺+ is not collapsed, becoming the new ω2 . Hence we apply Example 2 at the next stage to collapse ̺(ω1 ) = the ω1 -th successor of ̺ to ω1 . We continue in this fashion. We define an iteration hBi | i ≤ κi and a sequence h̺i | i ≤ κi as follows: ̺0 = ω1 , B0 = 2. (ω ) ̺i+1 = ̺i 1 and Bi+1 is constructed using Example 2 so as to collapse all regular τ ∈ (ω1 , ̺i+1 ) without collapsing ̺+ i+1 . For limit λ we proceed as follows: Case 1 λ has cofinality ω or has acquired cofinality ω at an earlier stage (i.e. cf(λ) < λ ∧ cf(λ) 6= ω1 in V). By essentially the above construction we form a limit Bλ which collapses ̺λ = (ω1 )  without collapsing ̺+ . sup ̺i i<λ Case 2 Case 1 fails. We set ̺λ = sup ̺i and let Bλ be the direct limit of hBi | i < λi. If cf(λ) = ω1 in i<λ V, then ̺+ λ becomes the new ω2 . Otherwise λ = ̺λ is inaccessible. Using the fact that we took the direct limit stationarily often below λ it follows that Bλ satisfies the λ-chain condition. Hence λ is the new ω2 . ⋆⋆⋆⋆⋆ By induction on i we verify that Bi is subcomplete for i ≤ κ, using Chapter 4, Theorem 4 for Case 2 above. We stress, however, that in order to carry out the July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in jensen 93 Examples induction we must also verify many other properties of the Bi which have not been dealt with here. These include some strong symmetry properties. Given that GCH holds below κ, we can modify the above construction by making selective regular τ ∈ (ω1 , κ) ω1 -cofinal. The set of such τ can be chosen arbitrarily in advance. Hence: Theorem Let κ be inaccessible. Let GCH hold below κ. Let A ⊂ κ. There is a set of conditions P ⊂ Vκ s.t. whenever G is P-generic, then in V[G] we have: • κ is ω2 . • If τ ∈ (ω1 , κ) is regular in V, then cf(τ ) =  ω1 if τ ∈ A, ω if not. July 21, 2012 15:2 94 World Scientific Book - 9.75in x 6.5in Subcomplete Forcing and L-Forcing jensen July 21, 2012 15:2 World Scientific Book - 9.75in x 6.5in Bibliography [ASS] J. Barwise. Admissible Sets and Structures, Perspectives in Math. Logic Vol. 7, Springer Verlag, 1976. [NA] H. Friedman, R. Jensen. A Note on Admissible Ordinals, in: The Syntax and Semantics of Infinitary Languages. Springer Lecture Notes in Math. Vol. 72, 1968. [PR] R. Jensen, C. Karp. Primitive Recursive Set Functions, in Axiomatic Set Theory, AMS Proceedings of Symposia in Pure Math. Vol. XIII, Part 1, 1971. [PF] S. Shelah. Proper and Improper Forcing, Perspectives in Math. Logic, Springer Verlag, 1998. [AS] R. Jensen: Admissible Sets*. [LF] R. Jensen. L-Forcing* [SPSC] R. Jensen. Subproper and Subcomplete Forcing* [ENP] R. Jensen. The Extended Namba Problem* [ITSC] R. Jensen. Iteration Theorems for Subcomplete and Related Forcings* [DSP] R. Jensen. Dee-Subproper Forcing* [FA] R. Jensen. Forcing Axioms Compatible with CH* * These handwritten notes can be downloaded from http://www.mathematik.hu-berlin/de/∼raesch/org/jensen.html (or enter ’Ronald B. Jensen’ in Google). 95 jensen