Papers by William Mitchell
Mathematical Research Letters, 1995
Theorem 0.1. Suppose that there is no inner model with a Woodin cardinal. Suppose that Ω is a mea... more Theorem 0.1. Suppose that there is no inner model with a Woodin cardinal. Suppose that Ω is a measurable cardinal. Let K be the Steel core model as computed in V Ω. Let κ ≥ ω 2 and λ = (κ +) K. Then cf(λ) ≥ card(κ). The main result of [MiSchSt] is that Theorem 0.1 holds under the additional assumption that card(κ) is countably closed. But often, in applications, countable closure is not available. Theorem 0.1 also builds on the earlier covering theorems of Jensen, Dodd and Jensen, and Mitchell; some of the relevant papers are [DeJe], [DoJe1], [DoJe2], [Mi1], [Mi2], and [Je]. The results for smaller core models do not require the existence of a measurable cardinal; it is not known if the large cardinal hypothesis on Ω can be eliminated completely from Theorem 0.1 (see [Sch2]). In this paper, we outline a proof of Theorem 0.1. By K c , we mean Steel's background certified core model. We shall reduce what we must prove to some iterability properties for K c (labeled "facts" in the proof). In turn, Steel has shown that K c is sufficiently iterable, using the methods in [St, §9]. The proof of Theorem 0.1 is very closely tied to the proof in [MiSchSt], to which we shall refer freely.
arXiv (Cornell University), Jul 26, 1995
We prove several results giving lower bounds for the large cardinal strength of a failure of the ... more We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem. Suppose κ is a singular strong limit cardinal and 2 κ ≥ λ where λ is not the successor of a cardinal of cofinality at most κ. If cf(κ) > ω then it follows that o(κ) ≥ λ, and if cf(κ) = ω then either o(κ) ≥ λ or { α : K |= o(α) ≥ α +n } is cofinal in κ for each n ∈ ω. We also prove several results which extend or are related to this result, notably Theorem. If 2 ω < ℵ ω and 2 ℵω > ℵ ω 1 then there is a sharp for a model with a strong cardinal. * Some of the results were obtained while Gitik was visiting Los Angeles in Fall 1991. He would like to thank A. Kechris, D. Martin and J. Steel for their hospitality.
Journal of Mathematical Logic
Assume that there is no transitive class model of [Formula: see text] with a Woodin cardinal. Let... more Assume that there is no transitive class model of [Formula: see text] with a Woodin cardinal. Let [Formula: see text] be a singular ordinal such that [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] is a regular cardinal in [Formula: see text]. Then [Formula: see text] is a measurable cardinal in [Formula: see text]. Moreover, if [Formula: see text], then [Formula: see text].
We answer a question of Shelah by showing that it is consistent that every set of ordinals of cof... more We answer a question of Shelah by showing that it is consistent that every set of ordinals of cofinality omega_1 in I[omega_2] is nonstationary if and only if it is consistent that that there is a kappa^+ Mahlo cardinal kappa.
Transactions of the American Mathematical Society, 2008
We answer a question of Shelah by showing that it is consistent that every member of I [ ω 2 ] ∩ ... more We answer a question of Shelah by showing that it is consistent that every member of I [ ω 2 ] ∩ Cof ( ω 1 ) I[\omega _2]\cap \operatorname {Cof}(\omega _1) is nonstationary if and only if it is consistent that there is a κ + \kappa ^+ -Mahlo cardinal κ \kappa .
Handbook of the History of Logic, 2012
This chapter is an introduction to the highly technical theory of inner models for large cardinal... more This chapter is an introduction to the highly technical theory of inner models for large cardinals. We present the fundamental concepts and ideas of the theory and state, mostly without a proof (or giving an outline of a proof) some significant results. There are two major themes in the theory of inner models. One is that with a given large cardinal property one can associate a minimal inner model of ZFC for that property. An example is the model L[U ] for a measurable cardinal. The other is the construction of core models for large cardinals. These generalize the Dodd-Jensen core model K that we describe in some detail. K is an inner model of ZFC that satisfies GCH and either the Covering Theorem holds for K (see below) or L[U ] exists. Definition 35.1. Let M be an inner model of ZFC. We say that the Covering Theorem holds for M if for every uncountable set X of ordinals there exists some Y ∈ M such that |Y | = |X|.
Hamkins and Löwe asked whether there can be a model N of set theory with the property that N ≡ N[... more Hamkins and Löwe asked whether there can be a model N of set theory with the property that N ≡ N[H] whenever H is a generic collapse of a cardinal of N onto ω. We obtain a lower bound, a cardinal κ with a κ +-repeat point, for the consistency of such a model. We do not know how to construct such a model, under any assumption. We do construct, from a cardinal κ with o(κ) = κ +, a model N which satisfies the desired condition when H is the collapse of any successor cardinal. We also give a much weaker lower bound for this property. Joel Hamkins and Benedikt Löwe have asked, in connection with results reported in [1], whether there can be a model N of ZFC set theory such that N[H] ≡ N whenever H is the generic collapse of any cardinal onto ω. This note gives some partial results related to this question. In the positive direction we have the following partial result: Theorem 1. Suppose there is a cardinal κ with o(κ) = κ +.
Transactions of the American Mathematical Society, 2001
We prove that o(κ) = κ is sufficient to construct a model V [C] in which κ is measurable and C is... more We prove that o(κ) = κ is sufficient to construct a model V [C] in which κ is measurable and C is a closed and unbounded subset of κ containing only inaccessible cardinals of V. Gitik proved that o(κ) = κ is necessary. We also calculate the consistency strength of the existence of such a set C together with the assumption that κ is Mahlo, weakly compact, or Ramsey. In addition we consider the possibility of having the set C generate the closed unbounded ultrafilter of V while κ remains measurable, and show that Radin forcing, which requires a weak repeat point, cannot be improved on.
Archive for Mathematical Logic, 2017
This paper is dedicated to the memory of Rich Laver and Jim Baumgartner, who I treasured as frien... more This paper is dedicated to the memory of Rich Laver and Jim Baumgartner, who I treasured as friends, colleagues and exemplars since we were graduate students together.
Transactions of the American Mathematical Society, 2009
We answer a question of Shelah by showing that it is consistent that every member of I[ω2] ∩ Cof(... more We answer a question of Shelah by showing that it is consistent that every member of I[ω2] ∩ Cof(ω1) is nonstationary if and only if it is consistent that that there is a κ +-Mahlo cardinal κ.
Handbook of Set Theory, 2009
Transactions of the American Mathematical Society, 1991
The main result of this paper is a characterization of singular cardinals in terms of the core mo... more The main result of this paper is a characterization of singular cardinals in terms of the core model, assuming that there is no model of 3/c o(k) = k++. This characterization is used to prove a result in infinitary Ramsey theory. In the course of the proof we develop a simplified statement of the covering lemma for sequences of measures which avoids the use of mice. We believe that this development will be capable of isolating almost all applications of the covering lemma from the detailed structure of the core model.
Journal of Symbolic Logic, 1999
We show that if there is no inner model with a Woodin cardinal and the Steel core model K exists,... more We show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ5-Erdős in K.In the absence of the Steel core model K we prove the same conclusion for any model L[] such that either V = L[] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is a generic extension of L[].The proof includes one lemma of independent interest: If V = L[A], where A ⊂ κ and κ is regular, then Lκ[A] is a Jónsson algebra. The proof of this result. Lemma 2.5, is very short and entirely elementary.
Journal of Symbolic Logic, 1998
If there is no inner model with a cardinal κ such that ο(κ) = κ++ then the set K ∩ Hω1 is definab... more If there is no inner model with a cardinal κ such that ο(κ) = κ++ then the set K ∩ Hω1 is definable over Hω1 by a Δ4 formula, and the set of countable initial segments of the core model is definable over Hω1 by a Π3 formula. We show that if there is an inner model with infinitely many measurable cardinals then there is a model in which is not definable Σ3 by any Σ3 formula, and K ∩ Hω1 is not definable by any boolean combination of Σ3 formulas.
Journal of Pure and Applied Algebra, 1972
Much work has been done with'the Elementary Theory of the Category of Sets (ETCS) introduced by L... more Much work has been done with'the Elementary Theory of the Category of Sets (ETCS) introduced by Lawvere in (21, particu!ariy since its reformulation by lawvere and Tierney I1.31. This work has been done under the assumption that a model of 'this theory is actually. in some sense. 3 category of sets. Our work is based on a subsystem of ETCS, the axioms for 3 Baolcan topos (BT). In Section 1, we define precisely the systenrs involved and give some basic constructions. Then in Section Z we define, for an arbitrary Boolean topos E, a Boolean model A{(E) and show, assuming that E satistItcs a weak axiom of choice (that the subobjects of I form a set of generators). that M(E) is a model da weak systetn %O of set theory. A stronger system 2, of set theory is also defined, and we show that if E satisfies brought to you by CORE View metadata, citation and similar papers at core.ac.uk
Journal of Pure and Applied Algebra, 1973
Annals of Pure and Applied Logic, 1996
We prove several results giving lower bounds for the large cardinal strength of a failure of the ... more We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem. Suppose κ is a singular strong limit cardinal and 2 κ ≥ λ where λ is not the successor of a cardinal of cofinality at most κ. If cf(κ) > ω then it follows that o(κ) ≥ λ, and if cf(κ) = ω then either o(κ) ≥ λ or { α : K |= o(α) ≥ α +n } is cofinal in κ for each n ∈ ω. We also prove several results which extend or are related to this result, notably Theorem. If 2 ω < ℵ ω and 2 ℵω > ℵ ω 1 then there is a sharp for a model with a strong cardinal. * Some of the results were obtained while Gitik was visiting Los Angeles in Fall 1991. He would like to thank A. Kechris, D. Martin and J. Steel for their hospitality.
Annals of Pure and Applied Logic, 2006
We give a short proof of a lemma which generalizes both the main lemma from the original construc... more We give a short proof of a lemma which generalizes both the main lemma from the original construction in the author's thesis of a model with no ω2-Aronszajn trees, and also the "Key Lemma" in Hamkins's gap forcing theorems. The new lemma directly yields Hamkins's newer lemma stating that certain forcing notions have the approximation property. According to Hamkins [2], a partial ordering P satisfies the δ-approximation property if, whenever A ∈ V P is a subset of an ordinal µ in V P such that A ∩ x ∈ V for each x ∈ ([µ] <δ) V , we have A ∈ V. In [2, Lemma 13] he proves the following lemma for the case when δ is a successor cardinal: Lemma 1. Suppose that δ is a regular cardinal, P * Q is a forcing in which P is nontrivial and |P | < δ, and PQ is <δ-strategically closed. Then P * Q has the δ-approximation property. This is a generalization of the "Key Lemma" of Hamkins's gap forcing theorems [1, 2]. Hamkins proves Lemma 1 from these key lemmas, which in turn are proved using a tree argument which is quite similar to the proof of the main lemma in the original construction [3] of a model with no ω 1-Aronszajn tree from a weakly compact cardinal κ. That model is constructed using a forcing which can be described in modern terms as a forcing iteration (P ν ,Q ν) : ν < κ of length κ. If ν < κ is a regular cardinal then Q ν is the forcing to add a Cohen real, andQ ν+1 is a name for the forcing to collapse ν onto ω 1 ; the forcing Q ν is trivial for all other ordinals ν < κ. The iteration uses finite support for the Cohen reals and countable support for the collapses. The main lemma of [3] states:
The Journal of Symbolic Logic, 2003
We reprove Gitik's theorem that if the GCH holds and o(κ) =κ+ 1 then there is a generic exten... more We reprove Gitik's theorem that if the GCH holds and o(κ) =κ+ 1 then there is a generic extension in whichκis still measurable and there is a closed unbounded subsetCofκsuch that every ν ∈Cis inaccessible in the ground model.Unlike the forcing used by Gitik, the iterated forcing ℛλ+1used in this paper has the property that ifλis a cardinal less thenκthen ℛλ+1can be factored inVas ℛκ+1= ℛλ+1× ℛλ+1,κwhere ∣ℛλ+1∣ ≤λ+and ℛλ+1,κdoes not add any new subsets ofλ.
Uploads
Papers by William Mitchell