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Unformatted text preview: uskal’s Theorem. Here is a modified view: impredicativity cannot be used for proving normal mathematical theorems that involve only finite objects. Here is a refutation of this. THEOREM 17. Let T be a sufficiently tall rooted finite tree of bounded valence (splitting). There is an inf preserving embedding of some truncation of T into a taller truncation of T which sends the highest vertices of the former into the highest vertices of the latter. Results from math logic again show that there is no predicative proof of this finite version of Kruskal’s Theorem. 9. ZERMELO SET THEORY. Zermelo set theory with the axiom of choice, ZC, is a very powerful fragment of the usual axioms and rules of mathematics (ZFC), and is far more than what is needed to formalize nearly all of existing normal mathematics. ZC consists of the axioms of exten-sionality, pairing, union, separation (comprehension), infinity, power set, and choice. We now give an example of a uniformization theorem from normal real analysis that cannot be proved in ZC. It can, however, be proved in ZFC, using the Replacement axiom. THEOREM 18. (using D.A. Martin). Let E be a Borel measurable subset of the ordinary unit square which is symmetric about the diagonal. Then E contains or is disjoint from the graph of a Borel measurable function from the unit interval into itself. 10. ZFC AND BEYOND. 14 Are there examples of discrete or even finite normal mathematics which cannot be carried out within the usual axioms and rules of mathematics as formalized by ZFC? This question naturally arises since even ZC is overkill for nearly all normal mathematical contexts. There is ongoing work suggesting that not only are there such examples, but that there is a new thematic subject which cuts across nearly all mathematical contexts, readily digestible at the undergraduate mathematics level, but which can be properly carried out with and only with the use of certain previously proposed new axioms for mathematics going under the name of “large cardinal axioms.” However, it would be premature for me to report on this work with any specificity at this important gathering, and so I will end this lecture at this time. Thank you very much....
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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