applctoborelfcns

applctoborelfcns - { PAGE } APPLICATIONS OF LARGE CARDINALS...

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Unformatted text preview: { PAGE } APPLICATIONS OF LARGE CARDINALS TO BOREL FUNCTIONS by Harvey M. Friedman Department of Mathematics Ohio State University friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman November 25, 1997 NOTE: This is work in progress. No proofs are presented. Results are still being checked. Let R be the set of all real numbers, and let CS(R) be the space of all nonempty countable subsets of R. The space CS(R) has a unique Borel structure in the following sense. Note that there is a natural mapping from R onto CS(R}; namely, taking ranges. We can combine this with any Borel bijection from R onto R in order to get a preferred surjection F:R f CS(R). In what sense is this preferred? Consider the following property * on F:R f CS(R): i) F is onto; ii) {(x,y1,y2,...): F(x) = F(y1) F(y2) ...} is a Borel measurable subset of R . By way of background, we have the following: THEOREM 1. Let F,G:R f CS(R) have property *. Then G is the result of composing F with a Borel permutation of R....
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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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applctoborelfcns - { PAGE } APPLICATIONS OF LARGE CARDINALS...

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