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Inaccessibility in Constructive Set Theory and Type Theory
, 1998
"... This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and MartinLof's intuitionistic theory of types. This paper treats Mahlo&apos ..."
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Cited by 20 (5 self)
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This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory (CZF) and MartinLof's intuitionistic theory of types. This paper treats Mahlo's numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders. Finally the theorems of that extension of CZF are interpreted in an extension of MartinLof's intuitionistic theory of types by a universe. 1 Prefatory and historical remarks The paper is organized as follows: After recalling Mahlo's numbers and relating the history of universes in MartinLof type theory in section 1, we study notions of inaccessibility in the context of Aczel's constructive set theo...
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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Cited by 11 (2 self)
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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.
The Higher Infinite in Proof Theory
 LOGIC COLLOQUIUM '95. LECTURE NOTES IN LOGIC
, 1995
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Metapredicative And Explicit Mahlo: A ProofTheoretic Perspective
"... After briefly discussing the concepts of predicativity, metapredicativity and impredicativity, we turn to the notion of Mahloness as it is treated in various contexts. Afterwards the ..."
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Cited by 3 (2 self)
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After briefly discussing the concepts of predicativity, metapredicativity and impredicativity, we turn to the notion of Mahloness as it is treated in various contexts. Afterwards the
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.
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