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Unformatted text preview: 1 THE INTERPRETATION OF SET THEORY IN PURE PREDICATION THEORY preliminary report by Harvey M. Friedman Department of Mathematics Ohio State University Columbus, Ohio 43210 August 20, 1997 [email protected] www.math.ohiostate.edu/~friedman/ NOTE: This paper is in preparation, and the only proofs present are in Part II, A and B. In fact, Godel gave an important model of pure predication, where he showed that restricted comprehension without parameters is valid, but where restricted comprehension with parameters is not (although this invalidity was not established until Cohen). This is the model based on ordinal definability in set theory. TABLE OF CONTENTS INTRODUCTION PART I. WITH EXTENSIONALITY. A. One subworld: ZF\P. B. Two subworlds: Indescribable and subtle cardinals. 1. Transitivity. 2. Ordinals and transfinite induction. 3. Finite ordinals, strong ordinals, and arithmetic. 4. Finite sequence codes. 5. Arithmetization, structures, and satisfaction relations. 6. Constructible universe structure. 7. Axioms of ZFC. 8. Indescribable caardinals, subtle cardinals, and ˛ models. C. Infinitely many distinguished subworlds: Subtle cardinals of finite order. D. The world predicate: Ramsey and measurable cardinals. E. The world predicate: Hypermeasurable cardinals. 2 F. Two subworlds: Elementary embeddings from V( l ) into V( l ). G. One subworld: Elementary embeddings incompatible with the axiom of choice. H. Infinitely many distinguished subworlds: Stronger axioms. PART II. WITHOUT EXTENSIONALITY. A. One subworld: ZF\P. B. Two subworlds: Indescribable and subtle cardinals. C. Infinitely many distinguished subworlds: Subtle cardinals of finite order. D. The world predicate: Ramsey and measurable cardinals. E. The world predicate: Hypermeasurable cardinals. F. Two subworlds: Elementary embeddings from V( l ) into V( l ). G. One subworld: Elementary embeddings incompatible with the axiom of choice. H. Infinitely many distinguished subworlds: Stronger axioms. PART III. CONCEPTUAL DISCUSSION. INTRODUCTION This paper was referred to in the Introductions to our papers [Fr97a], &The Axiomatization of Set Theory by Separation, Reducibility, and Comprehension,¡ and [Fr97b], &The Interpretation of Set Theory in Mathematical Predication Theory.¡ In [Fr97a], all systems considered include the axiom of Extensionality and made unrestricted use of parameters. Extensionality and unrestricted use of parameters is appropriate in the context of mathematical predication. In [Fr97b], all systems considered include unrestricted use of parameters, but not Extensionality. Unrestricted use of parameters is appropriate in the context of mathematical predication, and Extensionality is inappropriate....
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 Fall '08
 JOSHUA
 Math, Set Theory, Mathematical logic, Ordinal number, Transfinite induction, OW2, OW1

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