axiomsetthy - { PAGE } THE AXIOMATIZATION OF SET THEORY BY...

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Unformatted text preview: { PAGE } THE AXIOMATIZATION OF SET THEORY BY EXTENSIONALITY, SEPARATION, AND REDUCIBILITY preliminary report by Harvey M. Friedman Department of Mathematics Ohio State University Columbus, Ohio 43210 August 20, 1997 Revised October 28, 1997 friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ TABLE OF CONTENTS Introduction. 1. ZF\Foundation. 2. Indescribable and subtle cardinals. 3. Useful preliminaries in EST. 4. Measurable cardinals. 5. Elementary embeddings from V( a ) into V( b ). 6. ZF\Foundation again. 7. Elementary embeddings from V( a ) into V( a ). 8. Elementary embeddings from V into V. 9. Axiomatic elementary embeddings: huge cardinals and V( l ) into V( l ). 10. Axiomatic elementary embeddings: V into V. 11. Conceptual discussion. ADDENDUM - simplification of section 4 - October 28, 1997. INTRODUCTION All axiomatizations in sections 1,2,4-8 are in the language L( ,W) with just and the constant symbol W standing for a Subworld. Think of W as yesterday's world, and think of the quantifiers in the theory as ranging over today's world. The philosophy is that since the universe cannot be completed, every time we reflect on the universe and what we have reflected on previously, we obtain a larger universe. We lead off with a very simple reaxiomatization of ZF without Foundation using just three principles: Extensionality, Subworld separation, and Reducibility. We then use a modified form of Reducibility, and obtain an outright derivation of ZF\Foundation plus the existence of a large cardinal { PAGE } somewhere between indescribables and subtles, appropriately formulated in ZF without choice. The formulations are the same as the usual formulations if the axiom of choice is added. Also, ZF with these large cardinals corresponds to ZFC with these same large cardinals in one of several senses including equiconsistency. In sections 1-2, the axiomatizations prove that W is transitive. However, to get the bigger large cardinals in sections 4-8, we weaken subworld separation in a natural say to allow for W to be not transitive, and strengthen Reducibility in various ways. From the point of view of the technical development of current large cardinal theory, the idea behind W in sections 4-8 is that it is the range of an elementary embedding from a rank into a rank, and what we are axiomatizing in clear and simple terms is that image. In sections 9-10 we write down axioms that are much more directly motivated by the technical development of current large cardinal theory, by instead directly axiomatizing an elementary embedding via a unary function symbol. There is no subworld W. Instead we use epsilon, the unary function symbol representing the embedding, and equality. Equality is much more convenient here because of the function symbol, and we don't use equality as a primitive in the earlier sections....
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axiomsetthy - { PAGE } THE AXIOMATIZATION OF SET THEORY BY...

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