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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.
BARWISE: INFINITARY LOGIC AND ADMISSIBLE SETS
"... 1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The backandforth construction 3 ..."
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1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The backandforth construction 3
BERNAYS AND SET THEORY
"... Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Göd ..."
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Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the twovolume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of firstorder logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent reevaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higherorder reflection principles, and produced a stream of
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 is t ..."
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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].