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24
Equality and Extensionality in Automated HigherOrder Theorem Proving
, 1999
"... Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Hintikka Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Primitive Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Model Existenc ..."
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Cited by 14 (11 self)
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Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Hintikka Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Primitive Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Model Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Extensional HigherOrder Resolution: ER 42 4.1 A Review of HORES and ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Lifting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.5 Theorem Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5 Extensional HigherOrder Paramodulation: EP 57 5.1 A Naive and Incomplete Adaptation...
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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Cited by 11 (2 self)
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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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Cited by 8 (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 Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusi ..."
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Cited by 7 (2 self)
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
WE HOLD THESE TRUTHS TO BE SELFEVIDENT: BUT WHAT DO WE MEAN BY THAT?
"... Mathematicians at first distrusting the new ideas (Cantor made his first discoveries in 1873), then got used to them;... Waismann (1982, p. 102) Abstract. At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where pro ..."
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Mathematicians at first distrusting the new ideas (Cantor made his first discoveries in 1873), then got used to them;... Waismann (1982, p. 102) Abstract. At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of selfevidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are undermined at a crucial point, namely when selfevidence is supported by holistic and even pragmatic considerations. At the beginning of Die Grundlagen der Arithmetik (§2) (1884), Gottlob Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”, noting that “Euclid gives proofs of many things which anyone would concede him without question”. Frege sets himself the task of providing proofs of such basic arithmetic propositions as
Zermelo's WellOrdering Theorem in Type Theory
"... Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, wi ..."
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Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be wellordered. Zermelo's rst proof from 1904 is followed, with a simpli cation to avoid using comparability of wellorderings. The proof has been formalised in the system AgdaLight. 1
Cantor’s Countability Concept Contradicted
"... Cantor’s famous proof of the uncountability of real numbers is shown to apply to the set of natural numbers as well. Independently it is proved that the uncountability of the real numbers implies the uncountability of the rational numbers too. Finally it is shown that Cantor’s second diagonalizati ..."
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Cantor’s famous proof of the uncountability of real numbers is shown to apply to the set of natural numbers as well. Independently it is proved that the uncountability of the real numbers implies the uncountability of the rational numbers too. Finally it is shown that Cantor’s second diagonalization method is inapplicable at all because it lacks the diagonal. Hence, the conclusion that the cardinal number C of the continuum be larger than aleph0 is invalid. As a consequence, there remains no evidence for the existence of different infinities denoted by so−called transfinite cardinal numbers. The continuum hypothesis is not only undecidable but meaningless.
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