Transactions of the American Mathematical Society, Oct 1, 1993
Let A and B be two first order structures of the same vocabulary. We shall consider the Ehrenfeuc... more Let A and B be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraïssé-game of length ω 1 of A and B which we denote by G ω 1 (A, B). This game is like the ordinary Ehrenfeucht-Fraïssé-game of L ωω except that there are ω 1 moves. It is clear that G ω 1 (A, B) is determined if A and B are of cardinality ≤ ℵ 1. We prove the following results:
We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality... more We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality ℵ 2 such that the second player has a winning strategy in the Ehrenfeucht-Fraïssé-game of length ω 1 but there is no σ-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.
Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal µ, we ... more Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal µ, we prove the consistency of the existence of a wide µ +-Aronszajn tree, i.e. a tree of height and cardinality µ + with no branches of length µ + , into which every wide µ +-Aronszajn tree can be embedded.
Dependence logic, introduced by Väänänen, is the extension of first-order logic by atomic stateme... more Dependence logic, introduced by Väänänen, is the extension of first-order logic by atomic statements about (functional) dependencies of variables. An important feature of this logic is a model-theoretic semantics that, contrary to Tarski semantics, is not based on single assignments (mapping variables to elements of a structure) but on sets of assignments. Sets of assignments are called teams and the semantics is called team semantics. By focussing on independence rather than depencence, we have proposed a new logic, called independence logic, based on atomic formulae x ⊥z y which intuitively say that the variables x are independent from the variables y whenever the variables z are kept constant. We show that x ⊥z y gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We contrast this with dependence logic and show that independence logic has strictly more expressive power. Further, we will discuss game-theoretic semantics, expressive power, and complexity of dependence and independence logic.
The purpose of this paper is to examine the structural complexity of the sublogic relation betwee... more The purpose of this paper is to examine the structural complexity of the sublogic relation between abstract logics. Let ~(") denote n 'h order logic. Then cp(,) is a proper subtogic of A °("÷ 1~ for each n < co, and we have the chain ~(1)<~(2)< ... <~cp(,)< .... The question naturally arises, what other kinds of chains or partial orderings we can have among sufficiently regular abstract logics. Can one have a family of logics ordered in the order type of the rationals (or perhaps the reals)? In Chapter 2 we prove that any distributive lattice, and therefore any partial ordering, can be embedded into the sublogic relation among what we call normal logics. Chapter 3 is devoted to a sublogic relation defined using PC-classes rather than elementary classes. In the last chapter we restrict ourselves to the very special logics of the form ~(Q1,-, Q,)-The situation becomes more problematic but we can still prove that any countable partial ordering is embeddable into the sublogic relation of these logics. We confine ourselves to single-sorted structures, but many of the results are equally true of many-sorted structures.
A theory ~,f Boolean ~alued models for generalized quanliliels i:. dexelopcd ~ilh a ~pccial empha... more A theory ~,f Boolean ~alued models for generalized quanliliels i:. dexelopcd ~ilh a ~pccial emphasis on he Hfirtig-quantilier. As an application a Boolean exicnsion is obt;dned in xxhich the decision problem of the H~irlig-quantilim is ..~. lntroductie n Second-c, rder logic L n (with quantification over setsl is reputedly lacking model theoretic re ;ults. This situation had led to a search for well-behaved axiomatizablc fragments of L n, and indeed, many such interesting fragments have been found (see e.g. [8_' and [13]L The purpose of the present paper is to investigate those fragments o>: L n which fall well outside the category of these nice axiomatizable fragments. The simplest non-axiomatizable logic is LQ~,. where O.;A(x)<-~{aIAIo}} is infnite. There is, ho'vevcr, a straightforxvard h~tinitary formal system for l.O,+. E\cn the much strong,:r logic I.W.v.here Wx,:Atx. y)<-~ {~a. l'li .411~. hi} well-orders its field. permits a syntactical characlerization of validity ~sce [5]k No such results are known for LI. where lxy/dxIB(y~ "~+ card{t~ i A{a~} = caldII, ] B(bl}. This is tile Hiirtig-quantifier and is also denoted by O~. Ill the unix'erse of constructible sets LI is as powerful as L u itself, but there are models of set * The paper is based on a part of the author's Ph.D.-lhesis a~ Manche:.tcr t.J~fi'.er:,it). P-)77, The author wishes tc e~press here his gratitude to his stlpervisor. P. H. G. Aczel. for the help mtd encotlragenteill g'~.'ell during lhc preparation of the thesis, The ll,,esis and paper x~crc prepared v, hil¢ the aulhor was ti:lancially supporled b*, (.)sk. I:llltlltllC]l l~'olmdat!on.
Trees are natural generalizations of ordinals and this is especially apparent when one tries to ÿ... more Trees are natural generalizations of ordinals and this is especially apparent when one tries to ÿnd an uncountable analogue of the concept of the Scott-rank of a countable structure. The purpose of this paper is to introduce new methods in the study of an ordering between trees whose analogue is the usual ordering between ordinals. For example, one of the methods is the tree-analogue of the successor operation on the ordinals.
We introduce an atomic formula y ⊥ x z intuitively saying that the variables y are independent fr... more We introduce an atomic formula y ⊥ x z intuitively saying that the variables y are independent from the variables z if the variables x are kept constant. We contrast this with dependence logic D [5] based on the atomic formula =(x, y), actually a special case of y ⊥ x z, saying that the variables y are totally determined by the variables x. We show that y ⊥ x z gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We show that y ⊥ x z can be used to give partially ordered quantifiers and IF-logic a compositional interpretation without some of the shortcomings related to so called signaling that interpretations using =(x, y) have.
We make use of some observations on the core model, for example assuming V = L[E], that if there ... more We make use of some observations on the core model, for example assuming V = L[E], that if there is no inner model with a Woodin cardinal, and M is an inner model with the same cardinals as V , then V = M. We conclude in this latter situation that "x = P(y)" is Σ 1 (Card) where Card is a predicate true of just the infinite cardinals. It is known that this implies the validities of second order logic are reducible to V I the set of validities of the Härtig quantifier logic. We draw some further conclusions on the Löwenheim number, ℓ I of the latter logic: that if no L[E] model has a cardinal strong up to an ℵ-fixed point, and ℓ I is less than the least weakly inaccessible δ, then (i) ℓ I is a limit of measurable cardinals of K; (ii) the Weak Covering Lemma holds at δ.
Informally speaking, the categoricity of an axiom system means that its nonlogical symbols have o... more Informally speaking, the categoricity of an axiom system means that its nonlogical symbols have only one possible interpretation that renders the axioms true. Although non-categoricity has become ubiquitous in the second half of the 20th century whether one looks at number theory, geometry or analysis, the first axiomatizations of such mathematical theories by Dedekind, Hilbert, Huntington, Peano and Veblen were indeed categorical. A common resolution of the difference between the earlier categorical axiomatizations and the more modern non-categorical axiomatizations is that the latter derive their non-categoricity from Skolem's Paradox and Gödel's Incompleteness Theorems, while the former, being second order, suffer from a heavy reliance on metatheory, where the Skolem-Gödel phenomenon re-emerges. Using second order meta-theory to avoid non-categoricity of the metatheory would only seem to lead to an infinite regress. In this paper we maintain that internal categoricity breaks this traditional picture. It applies to both first and second order axiomatizations, although in the first order case we have so far only examples. It does not depend on the meta-theory in a way that would lead to an infinite regress. And it covers the classical categoricity results of early researchers. In the first order case it is weaker than categoricity itself, and in the second order case stronger. We give arguments suggesting that internal categoricity is the "right" concept of categoricity. * I am indebted to John Baldwin for his critical reading and comments on an earlier manuscript of this paper, and to Roman Kossak for a discussion on models of arithmetic in relation to this paper. I am also indebted to the referees for valuable remarks.
model theory is the attempt to systematize the study of logics by studying the relationships betw... more model theory is the attempt to systematize the study of logics by studying the relationships between them and between various of their properties. The perspective taken in abstract model theory is discussed in Section 2 of Chapter I. The basic definitions and results of the subject were presented in Part A. Other results are scattered throughout the book. This final part of the book is devoted to more advanced topics in abstract model theory.
We consider the logic L q, introduced by Thomason [8], which is obtained from L ωω by adding mona... more We consider the logic L q, introduced by Thomason [8], which is obtained from L ωω by adding monadic quantifier variables. Thus if A 1,…,A n are formulae of L q, x 1,…,x n individual variables and Q a quantifier variable of type , then $$Q{x_1} \ldots {x_n}{A_1}({x_1}) \ldots {A_n}({x_n})$$ is a formula of L q. An interpretation of L q is obtained from an interpretation of L ωω by assigning to the quantifier variables mondaic generalized quantifiers in the sense of Lindstrom [6]. Thus a quantifier variable Q of type is assigned in a given domain I a class Q of structures of the form such that X i⊆I(i=1…n) and every isomorphic copy of a structure in Q is also in Q. In this assignment $$ \left\langle {I,R_1 , \ldots ,R_m } \right\rangle \left| { = Qx_1 \ldots x_m A_1 \left( {x_1 ,y} \right) \ldots A_n \left( {x_n ,y} \right)\left[ a \right]} \right. $$ iff $$ \left\langle {I,X_1 , \ldots ,X_n } \right\rangle \in 2 $$ where $$X_i = \left\{ {b \in I\left| {\left\langle {I,R_1 , \ldots ,R_m } \right\rangle \left| { = A_i \left( {x_i ,y} \right)\left[ {ba} \right]} \right.} \right.} \right\}.$$
We consider team semantics for propositional logic, continuing [34]. In team semantics the truth ... more We consider team semantics for propositional logic, continuing [34]. In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility to give meaning to concepts such as dependence, independence and inclusion. We associate with every formula φ based on finitely many propositional variables the set φ of teams that satisfy φ. We define a full propositional team logic in which every set of teams is definable as φ for suitable φ. This requires going beyond the logical operations of classical propositional logic. We exhibit a hierarchy of logics between the smallest, viz. classical propositional logic, and the full propositional team logic. We characterize these different logics in several ways: first syntactically by their logical operations, and then semantically by the kind of sets of teams they are capable of defining. In several important cases we are able to find complete axiomatizations for these logics.
A fundamental notion in a large part of mathematics is the notion of equicardinality. The languag... more A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Härtig quantifier is, roughly speaking, a first-order language in which the notion of equicardinality is expressible. Thus this language, denoted by LI, is in some sense very natural and has in consequence special interest. Properties of LI are studied in many papers. In [BF, Chapter VI] there is a short survey of some known results about LI. We feel that a more extensive exposition of these results is needed.The aim of this paper is to give an overview of the present knowledge about the language LI and list a selection of open problems concerning it.After the Introduction (§1), in §§2 and 3 we give the fundamental results about LI. In §4 the known model-theoretic properties are discussed. The next section is devoted to properties of mathematical theories in LI. In §6 the spectra of sentences of LI are discussed, and §7 is devoted to properties of LI which depend on set-theoretic assumptions. The paper finishes with a list of open problem and an extensive bibliography. The bibliography contains not only papers we refer to but also all papers known to us containing results about the language with Härtig quantifier.Contents. §1. Introduction. §2. Preliminaries. §3. Basic results. §4. Model-theoretic properties of LI. §5. Decidability of theories with I. §6. Spectra of LI-sentences. §7. Independence results. §8. What is not yet known about LI. Bibliography.
Transactions of the American Mathematical Society, Oct 1, 1993
Let A and B be two first order structures of the same vocabulary. We shall consider the Ehrenfeuc... more Let A and B be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraïssé-game of length ω 1 of A and B which we denote by G ω 1 (A, B). This game is like the ordinary Ehrenfeucht-Fraïssé-game of L ωω except that there are ω 1 moves. It is clear that G ω 1 (A, B) is determined if A and B are of cardinality ≤ ℵ 1. We prove the following results:
We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality... more We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality ℵ 2 such that the second player has a winning strategy in the Ehrenfeucht-Fraïssé-game of length ω 1 but there is no σ-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.
Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal µ, we ... more Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal µ, we prove the consistency of the existence of a wide µ +-Aronszajn tree, i.e. a tree of height and cardinality µ + with no branches of length µ + , into which every wide µ +-Aronszajn tree can be embedded.
Dependence logic, introduced by Väänänen, is the extension of first-order logic by atomic stateme... more Dependence logic, introduced by Väänänen, is the extension of first-order logic by atomic statements about (functional) dependencies of variables. An important feature of this logic is a model-theoretic semantics that, contrary to Tarski semantics, is not based on single assignments (mapping variables to elements of a structure) but on sets of assignments. Sets of assignments are called teams and the semantics is called team semantics. By focussing on independence rather than depencence, we have proposed a new logic, called independence logic, based on atomic formulae x ⊥z y which intuitively say that the variables x are independent from the variables y whenever the variables z are kept constant. We show that x ⊥z y gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We contrast this with dependence logic and show that independence logic has strictly more expressive power. Further, we will discuss game-theoretic semantics, expressive power, and complexity of dependence and independence logic.
The purpose of this paper is to examine the structural complexity of the sublogic relation betwee... more The purpose of this paper is to examine the structural complexity of the sublogic relation between abstract logics. Let ~(") denote n 'h order logic. Then cp(,) is a proper subtogic of A °("÷ 1~ for each n < co, and we have the chain ~(1)<~(2)< ... <~cp(,)< .... The question naturally arises, what other kinds of chains or partial orderings we can have among sufficiently regular abstract logics. Can one have a family of logics ordered in the order type of the rationals (or perhaps the reals)? In Chapter 2 we prove that any distributive lattice, and therefore any partial ordering, can be embedded into the sublogic relation among what we call normal logics. Chapter 3 is devoted to a sublogic relation defined using PC-classes rather than elementary classes. In the last chapter we restrict ourselves to the very special logics of the form ~(Q1,-, Q,)-The situation becomes more problematic but we can still prove that any countable partial ordering is embeddable into the sublogic relation of these logics. We confine ourselves to single-sorted structures, but many of the results are equally true of many-sorted structures.
A theory ~,f Boolean ~alued models for generalized quanliliels i:. dexelopcd ~ilh a ~pccial empha... more A theory ~,f Boolean ~alued models for generalized quanliliels i:. dexelopcd ~ilh a ~pccial emphasis on he Hfirtig-quantilier. As an application a Boolean exicnsion is obt;dned in xxhich the decision problem of the H~irlig-quantilim is ..~. lntroductie n Second-c, rder logic L n (with quantification over setsl is reputedly lacking model theoretic re ;ults. This situation had led to a search for well-behaved axiomatizablc fragments of L n, and indeed, many such interesting fragments have been found (see e.g. [8_' and [13]L The purpose of the present paper is to investigate those fragments o>: L n which fall well outside the category of these nice axiomatizable fragments. The simplest non-axiomatizable logic is LQ~,. where O.;A(x)<-~{aIAIo}} is infnite. There is, ho'vevcr, a straightforxvard h~tinitary formal system for l.O,+. E\cn the much strong,:r logic I.W.v.here Wx,:Atx. y)<-~ {~a. l'li .411~. hi} well-orders its field. permits a syntactical characlerization of validity ~sce [5]k No such results are known for LI. where lxy/dxIB(y~ "~+ card{t~ i A{a~} = caldII, ] B(bl}. This is tile Hiirtig-quantifier and is also denoted by O~. Ill the unix'erse of constructible sets LI is as powerful as L u itself, but there are models of set * The paper is based on a part of the author's Ph.D.-lhesis a~ Manche:.tcr t.J~fi'.er:,it). P-)77, The author wishes tc e~press here his gratitude to his stlpervisor. P. H. G. Aczel. for the help mtd encotlragenteill g'~.'ell during lhc preparation of the thesis, The ll,,esis and paper x~crc prepared v, hil¢ the aulhor was ti:lancially supporled b*, (.)sk. I:llltlltllC]l l~'olmdat!on.
Trees are natural generalizations of ordinals and this is especially apparent when one tries to ÿ... more Trees are natural generalizations of ordinals and this is especially apparent when one tries to ÿnd an uncountable analogue of the concept of the Scott-rank of a countable structure. The purpose of this paper is to introduce new methods in the study of an ordering between trees whose analogue is the usual ordering between ordinals. For example, one of the methods is the tree-analogue of the successor operation on the ordinals.
We introduce an atomic formula y ⊥ x z intuitively saying that the variables y are independent fr... more We introduce an atomic formula y ⊥ x z intuitively saying that the variables y are independent from the variables z if the variables x are kept constant. We contrast this with dependence logic D [5] based on the atomic formula =(x, y), actually a special case of y ⊥ x z, saying that the variables y are totally determined by the variables x. We show that y ⊥ x z gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. We show that y ⊥ x z can be used to give partially ordered quantifiers and IF-logic a compositional interpretation without some of the shortcomings related to so called signaling that interpretations using =(x, y) have.
We make use of some observations on the core model, for example assuming V = L[E], that if there ... more We make use of some observations on the core model, for example assuming V = L[E], that if there is no inner model with a Woodin cardinal, and M is an inner model with the same cardinals as V , then V = M. We conclude in this latter situation that "x = P(y)" is Σ 1 (Card) where Card is a predicate true of just the infinite cardinals. It is known that this implies the validities of second order logic are reducible to V I the set of validities of the Härtig quantifier logic. We draw some further conclusions on the Löwenheim number, ℓ I of the latter logic: that if no L[E] model has a cardinal strong up to an ℵ-fixed point, and ℓ I is less than the least weakly inaccessible δ, then (i) ℓ I is a limit of measurable cardinals of K; (ii) the Weak Covering Lemma holds at δ.
Informally speaking, the categoricity of an axiom system means that its nonlogical symbols have o... more Informally speaking, the categoricity of an axiom system means that its nonlogical symbols have only one possible interpretation that renders the axioms true. Although non-categoricity has become ubiquitous in the second half of the 20th century whether one looks at number theory, geometry or analysis, the first axiomatizations of such mathematical theories by Dedekind, Hilbert, Huntington, Peano and Veblen were indeed categorical. A common resolution of the difference between the earlier categorical axiomatizations and the more modern non-categorical axiomatizations is that the latter derive their non-categoricity from Skolem's Paradox and Gödel's Incompleteness Theorems, while the former, being second order, suffer from a heavy reliance on metatheory, where the Skolem-Gödel phenomenon re-emerges. Using second order meta-theory to avoid non-categoricity of the metatheory would only seem to lead to an infinite regress. In this paper we maintain that internal categoricity breaks this traditional picture. It applies to both first and second order axiomatizations, although in the first order case we have so far only examples. It does not depend on the meta-theory in a way that would lead to an infinite regress. And it covers the classical categoricity results of early researchers. In the first order case it is weaker than categoricity itself, and in the second order case stronger. We give arguments suggesting that internal categoricity is the "right" concept of categoricity. * I am indebted to John Baldwin for his critical reading and comments on an earlier manuscript of this paper, and to Roman Kossak for a discussion on models of arithmetic in relation to this paper. I am also indebted to the referees for valuable remarks.
model theory is the attempt to systematize the study of logics by studying the relationships betw... more model theory is the attempt to systematize the study of logics by studying the relationships between them and between various of their properties. The perspective taken in abstract model theory is discussed in Section 2 of Chapter I. The basic definitions and results of the subject were presented in Part A. Other results are scattered throughout the book. This final part of the book is devoted to more advanced topics in abstract model theory.
We consider the logic L q, introduced by Thomason [8], which is obtained from L ωω by adding mona... more We consider the logic L q, introduced by Thomason [8], which is obtained from L ωω by adding monadic quantifier variables. Thus if A 1,…,A n are formulae of L q, x 1,…,x n individual variables and Q a quantifier variable of type , then $$Q{x_1} \ldots {x_n}{A_1}({x_1}) \ldots {A_n}({x_n})$$ is a formula of L q. An interpretation of L q is obtained from an interpretation of L ωω by assigning to the quantifier variables mondaic generalized quantifiers in the sense of Lindstrom [6]. Thus a quantifier variable Q of type is assigned in a given domain I a class Q of structures of the form such that X i⊆I(i=1…n) and every isomorphic copy of a structure in Q is also in Q. In this assignment $$ \left\langle {I,R_1 , \ldots ,R_m } \right\rangle \left| { = Qx_1 \ldots x_m A_1 \left( {x_1 ,y} \right) \ldots A_n \left( {x_n ,y} \right)\left[ a \right]} \right. $$ iff $$ \left\langle {I,X_1 , \ldots ,X_n } \right\rangle \in 2 $$ where $$X_i = \left\{ {b \in I\left| {\left\langle {I,R_1 , \ldots ,R_m } \right\rangle \left| { = A_i \left( {x_i ,y} \right)\left[ {ba} \right]} \right.} \right.} \right\}.$$
We consider team semantics for propositional logic, continuing [34]. In team semantics the truth ... more We consider team semantics for propositional logic, continuing [34]. In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility to give meaning to concepts such as dependence, independence and inclusion. We associate with every formula φ based on finitely many propositional variables the set φ of teams that satisfy φ. We define a full propositional team logic in which every set of teams is definable as φ for suitable φ. This requires going beyond the logical operations of classical propositional logic. We exhibit a hierarchy of logics between the smallest, viz. classical propositional logic, and the full propositional team logic. We characterize these different logics in several ways: first syntactically by their logical operations, and then semantically by the kind of sets of teams they are capable of defining. In several important cases we are able to find complete axiomatizations for these logics.
A fundamental notion in a large part of mathematics is the notion of equicardinality. The languag... more A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Härtig quantifier is, roughly speaking, a first-order language in which the notion of equicardinality is expressible. Thus this language, denoted by LI, is in some sense very natural and has in consequence special interest. Properties of LI are studied in many papers. In [BF, Chapter VI] there is a short survey of some known results about LI. We feel that a more extensive exposition of these results is needed.The aim of this paper is to give an overview of the present knowledge about the language LI and list a selection of open problems concerning it.After the Introduction (§1), in §§2 and 3 we give the fundamental results about LI. In §4 the known model-theoretic properties are discussed. The next section is devoted to properties of mathematical theories in LI. In §6 the spectra of sentences of LI are discussed, and §7 is devoted to properties of LI which depend on set-theoretic assumptions. The paper finishes with a list of open problem and an extensive bibliography. The bibliography contains not only papers we refer to but also all papers known to us containing results about the language with Härtig quantifier.Contents. §1. Introduction. §2. Preliminaries. §3. Basic results. §4. Model-theoretic properties of LI. §5. Decidability of theories with I. §6. Spectra of LI-sentences. §7. Independence results. §8. What is not yet known about LI. Bibliography.
Uploads
Papers by Jouko Väänänen