ConStThyAmst050610pdf

ConStThyAmst050610pdf - ASPECTS OF CONSTRUCTIVE SET THEORY...

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ASPECTS OF CONSTRUCTIVE SET THEORY AND BEYOND by Harvey M. Friedman Ohio State University www.math.ohio-state.edu/~friedman/ Set Theory, Classical and Constructive May 6, 2010
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WHAT ARE THESE THREE ASPECTS? 1. FORMALIZING CONSTRUCTIVE ANALYSIS. Provide a formal system that is a conservative extension of PA for Π 02 sentences, and even a conservative extension of HA, that supports the worry free smooth development of constructive analysis in the style of Errett Bishop. 2. FORMALIZING CLASSICAL ANALYSIS. Prove a formal system that is a conservative extension of PA, that supports the worry free smooth development of classical analysis. 3. STRONG INTUITIONISTIC ZF. Understand strong intuitionistic versions of ZF. 4. INTUITIONISTIC LARGE CARDINAL THEORY. Understand intuitionistic versions of large cardinal hypotheses.
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1. FORMALIZING CONSTRUCTIVE ANALYSIS We gave a foundation for Errett Bishop style constructive analysis in terms of intuitionistic set theory, in H. Friedman, Foundations for Constructive Analysis, Annals of Mathematics, 105 (1977), 1-28. In this paper, we show that any arithmetic sentence provable in the system B is provable in PA, and that B and PA prove the same Π 02 sentences. In another paper with a different purpose (discussed below), I wrote "In Friedman (1977), we presented a fragment B of Zermelo set theory with intuitionistic logic, and proved that any arithmetic sentence provable in B is provable in PA. (It is now known that B is a conservative extension of HA.)"
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1. FORMALIZING CONSTRUCTIVE ANALYSIS Unfortunately, I didn't give a reference, and I don't quite remember what that reference should be, or whether it was an unpublished observation of ours. In any case, I'm sure that this (sort of thing) must be embedded in the current literature. I think this was done using the formalized realizability method that Michael Beeson presented in his thesis to reprove a theorem of Goodman in his thesis, that HA is conservative over HA. The primitives of B are "being a natural number", "being a set", "y is the successor of x", and identity. The axioms of B are as follows. A. (Ontological axioms.) Every object is either a set or a natural number, but not both. 0 is a natural number. If x is the successor of y, then x,y are natural numbers. If x y then y is a set. If x = y then x is a set if and only if y is a set. B. (Equality axioms).
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1. FORMALIZING CONSTRUCTIVE ANALYSIS C. (Extensionality.) If two sets have the same elements then they are equal. D. (Successor axioms.) Any two successors of a number are equal. Any two numbers with the same successor are equal. 0 is not the successor of any number. E. (Infinity.) The set of all natural numbers exists. F. (Induction.) If a set contains 0 and is closed under successor, then it
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ConStThyAmst050610pdf - ASPECTS OF CONSTRUCTIVE SET THEORY...

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