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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 Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
A Hanf number for saturation and omission
"... Suppose t = (T, T1, p) is a triple of two countable theories T ⊆ T1 in vocabularies τ ⊂ τ1 and a τ1type p over the empty set. We show the Hanf number for the property: There is a model M1 of T1 which omits p, but M1 ↾ τ is saturated is essentially equal to the Löwenheim number of second order logic ..."
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Suppose t = (T, T1, p) is a triple of two countable theories T ⊆ T1 in vocabularies τ ⊂ τ1 and a τ1type p over the empty set. We show the Hanf number for the property: There is a model M1 of T1 which omits p, but M1 ↾ τ is saturated is essentially equal to the Löwenheim number of second order logic. In Section 4 we make exact computations of these Hanf numbers and note some interesting distinctions between ‘first order ’ and ‘second order quantification’. In particular, we show that if κ is uncountable, h 3 (Lω,ω(Q), κ) = h 3 (Lω1,ω, κ), where h3 is the ‘normal ’ notion of Hanf function (Definition 4.13.) Newelski asked in [New] whether it is possible to calculate the Hanf number of the following property PN. In a sense made precise in Theorem 0.2, we show the answer is no. In accordance with the original question, we focus on countable vocabularies for the first three sections. We deal with extensions to larger vocabularies in Section 4. Definition 0.1 We say M1  = t where t = (T, T1, p) is a triple of two theories in vocabularies τ ⊂ τ1, respectively, T ⊆ T1 and p is a τ1type over the empty set if M1
On Banach Spaces of Large Density Character and Unconditional Sequences
, 2002
"... Dedicated to the memory of S. Banach. Abstract. It is shown that any Banach space X of density dens(X) ≥ גω1 contains an (infinite) unconditional sequence. Here for any ordinal α the cardinal גα is given by induction: ג0 = ℵ0; גα+1 = exp(גα); גα = sup{גβ: β < α} if α is a limit ordinal, and ω1 i ..."
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Dedicated to the memory of S. Banach. Abstract. It is shown that any Banach space X of density dens(X) ≥ גω1 contains an (infinite) unconditional sequence. Here for any ordinal α the cardinal גα is given by induction: ג0 = ℵ0; גα+1 = exp(גα); גα = sup{גβ: β < α} if α is a limit ordinal, and ω1 is the first uncountable ordinal. Every Banach space of density dens(X) = τ ≥ ω1 contains an unconditional sequence of cardinality τ if and only if τ is a weakly compact cardinal. Other results that are concerned large cardinals and unconditional sequences are also presented. 1. Introduction. The known J. Lindenstrauss ’ problem [1] was: Whether every infinite dimensional Banach space contains an (infinite) unconditional sequence? It was successfully solved (in negative) by W.T. Gowers and B. Maurey [2].
SET THEORY FROM CANTOR TO COHEN
"... Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun int ..."
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Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The
© Birkhäuser Verlag, Basel, 2001
"... Full duality among graph algebras and flat graph algebras ..."
INFINITARY LANGUAGES by
"... We begin with the following quotation from Karp [1964]: My interest in infinitary logic dates back to a February day in 1956 when I remarked to my thesis supervisor, Professor Leon Henkin, that a particularly vexing problem would be so simple if only I could write a formula which would say x = 0 or ..."
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We begin with the following quotation from Karp [1964]: My interest in infinitary logic dates back to a February day in 1956 when I remarked to my thesis supervisor, Professor Leon Henkin, that a particularly vexing problem would be so simple if only I could write a formula which would say x = 0 or x = 1 or x = 2 etc. To my surprise, he replied, &quot;Well, go ahead.&quot;
BERNAYS AND SET THEORY
"... We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. ..."
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We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles.