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24
Does Mathematics Need New Axioms?
 American Mathematical Monthly
, 1999
"... this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called f ..."
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this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 1997
"... A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is ma ..."
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A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinaltheoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie "  the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency  technical results in pro...
The Tree Property
 Adv. Math
"... . We construct a model in which there are no @nAronszajn trees for any finite n 2, starting from a model with infinitely many supercompact cardinals. We also construct a model in which there is no ++ Aronszajn tree for a strong limit cardinal of cofinality !, starting from a model with a ..."
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. We construct a model in which there are no @nAronszajn trees for any finite n 2, starting from a model with infinitely many supercompact cardinals. We also construct a model in which there is no ++ Aronszajn tree for a strong limit cardinal of cofinality !, starting from a model with a supercompact cardinal and a weakly compact cardinal above it. 1. Introduction We will prove the following theorems. Theorem 1. If "ZFC + there exist infinitely many supercompact cardinals" is consistent, then "ZFC + there are no @nAronszajn trees for 2 n ! !" is also consistent. Theorem 2. If "ZFC + there exists a supercompact cardinal with a weakly compact cardinal above it" is consistent then "ZFC + there exists a strong limit cardinal of cofinality ! such that there are no ++ Aronszajn trees" is also consistent. We start by recalling the definition of "Aronszajn tree" and some related concepts. Definition 1.1. Let be regular. 1. A tree is a tree of height whose every lev...
Menas' Result is Best Possible
, 1995
"... Generalizing some earlier techniques due to the second author, we show that Menas’ theorem which states that the least cardinal κ which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not 2 κ supercompact is best possible. Using these same techniques, we ..."
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Generalizing some earlier techniques due to the second author, we show that Menas’ theorem which states that the least cardinal κ which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not 2 κ supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author.
On the singular cardinal hypothesis
 Trans. Amer. Math. Soc
, 1992
"... The Singular Cardinal Hypothesis (SCH) asserts that if κ is any singular strong limit cardinal then 2 κ = κ +. It is known to be consistent that the SCH fails: Prikry [Pr] obtains a model of ¬SCH from a model in which the GCH fails at a measurable cardinal κ, and Silver in turn (see [KM]) obtains th ..."
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The Singular Cardinal Hypothesis (SCH) asserts that if κ is any singular strong limit cardinal then 2 κ = κ +. It is known to be consistent that the SCH fails: Prikry [Pr] obtains a model of ¬SCH from a model in which the GCH fails at a measurable cardinal κ, and Silver in turn (see [KM]) obtains the failure of the GCH at a
Large Cardinal Properties of Small Cardinals
 In Set theory (Curacao
, 1998
"... Introduction The fact that small cardinals (for example @ 1 and @ 2 ) can consistently have properties similar to those of large cardinals (for example measurable or supercompact cardinals) is a recurring theme in set theory. In these notes I discuss three examples of this phenomenon; stationary re ..."
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Introduction The fact that small cardinals (for example @ 1 and @ 2 ) can consistently have properties similar to those of large cardinals (for example measurable or supercompact cardinals) is a recurring theme in set theory. In these notes I discuss three examples of this phenomenon; stationary reflection, saturated ideals and the tree property. These notes represent approximately the contents of a series of expository lectures given during the Set Theory meeting at CRM Barcelona in June 1996. None of the results discussed here is due to me unless I say so explicitly. I would like to express my thanks to Joan Bagaria and Adrian Mathias for organising a very enjoyable meeting. 1 2 Large cardinals and elementary embeddings We begin by reviewing the formulation of large cardinal properties in terms of elementary embeddings. See [40], [22] or [21] for more on this topic. We will write "j : V<F14.4
Identity Crises and Strong Compactness
, 1998
"... : Combining techniques of the rst author and Shelah with ideas of Magidor, we show how to get a model in which, for xed but arbitrary nite n, the rst n strongly compact cardinals 1 ; : : : ; n are so that i for i = 1; : : : ; n is both the i th measurable cardinal and + i supercompact. This ..."
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Cited by 3 (3 self)
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: Combining techniques of the rst author and Shelah with ideas of Magidor, we show how to get a model in which, for xed but arbitrary nite n, the rst n strongly compact cardinals 1 ; : : : ; n are so that i for i = 1; : : : ; n is both the i th measurable cardinal and + i supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah. Supported by the VolkswagenStiftung (RiPprogram at Oberwolfach). In addition, this research was partially supported by PSCCUNY Grant 667379. Supported by the VolkswagenStiftung (RiPprogram at Oberwolfach). In addition, this research was partially supported by NSF Grant DMS9703945. Both authors wish to express their gratitude to Menachem Magidor for his explanations to them given at the January 713, 1996 meeting in Set Theory held at the Mathematics Research Institute, Oberwolfach on his method of forcing to make the rst n measurable and strongly compact cardinals coincide,...
Amplification of Completely Bounded Operators and Tomiyama’s Slice Maps
, 2002
"... Let (M, N) be a pair of von Neumann algebras, or of dual operator spaces with at least one of them having property Sσ, and let Φ be an arbitrary completely bounded mapping on M. We present an explicit construction of an amplification of Φ to a completely bounded mapping on M⊗N. Our approach is based ..."
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Let (M, N) be a pair of von Neumann algebras, or of dual operator spaces with at least one of them having property Sσ, and let Φ be an arbitrary completely bounded mapping on M. We present an explicit construction of an amplification of Φ to a completely bounded mapping on M⊗N. Our approach is based on the concept of slice maps as introduced by Tomiyama, and makes use of the description of the predual of M⊗N given by Effros and Ruan in terms of the operator space projective tensor product (cf. [Eff–Rua 90], [Rua 92]). We further discuss several properties of an amplification in connection with the investigations made in [May–Neu–Wit 89], where the special case M = B(H) and N = B(K) has been considered (for Hilbert spaces H and K). We will then mainly focus on various applications, such as a remarkable purely algebraic characterization of w ∗continuity using amplifications, as well as a generalization of the socalled Ge–Kadison Lemma (in connection with the uniqueness problem of amplifications). Finally, our study will enable us to show that the essential assertion of the main result in [May–Neu–Wit 89] concerning completely bounded bimodule homomorphisms actually relies on a basic property of Tomiyama’s slice maps. 1
An Ordinal Analysis of Stability
 ARCHIVE FOR MATHEMATICAL LOGIC
, 2005
"... This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2 comprehension. On the settheoretic side 1 2 comprehension corresponds to KripkePlatek set theory, KP, plus 1 separation. The strength of the latter theory is encapsulated in the fact that it proves th ..."
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This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2 comprehension. On the settheoretic side 1 2 comprehension corresponds to KripkePlatek set theory, KP, plus 1 separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals such that, for all > , is stable, i.e. L is a 1 elementary substructure of L . The objective of this paper is to give an ordinal analysis of not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of 1 2 comprehension is greatly facilated by explicating certain simpler cases rst. This paper introduces an ordinal representation system based on indescribable cardinals which is then employed for determining an upper bound for the proof{ theoretic strength of the theory KPi + 8 9 is + stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.