Pi01120305 - 1 P01 INCOMPLETENESS by Harvey M Friedman...

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1 P 0 1 INCOMPLETENESS by Harvey M. Friedman December 3, 2005 "Beautiful" is a word used by mathematicians with a semi rigorous meaning. We give "arguably beautiful" explicitly Pi01 sentences independent of ZFC. See Proposition A from section 1, and variants. 1. P 0 1 INDEPENDENT STATEMENTS USING ANTICHAINS. We use [1,n] for the discrete interval {1,. ..,n}. Let A [1,n] k . We write A’ = [1,n] k \A. This treats [1,n] k as the ambient space. Let R [1,n] 2k . We define RA = R[A] = {y [1,n] k : ( $ x A)(R(x,y))}. We say that R is strictly dominating if and only if for all x,y [1,n] k , if R(x,y) then max(x) < max(y). We start with a basic finite fixed point theorem. THEOREM 1.1. For all k,n 1 and strictly dominating R [1,n] 2k , there exists A [1,n] k such that R[A’] = A. Furthermore, A [1,n] k is unique. We can obviously take complements, obtaining what we call the ‘complementation theorem’ for RA. THEOREM 1.2. For all k,n 1 and strictly dominating R [1,n] 2k , there exists A [1,n] k such that RA = A’. Furthermore, A [1,n] k is unique. For A [1,n] k and t 1, we write A\t = {x A: t is not a coordinate of x} = "A with t omitted". Here is a modification of Theorem 1.1 which we call the ‘complementation theorem’ for R[A\2 (8k)! -1].
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2 Here 2 (8k)! -1 is one less than 2 to the exponent (8k)!. This is a convenient decrement by 1 of a sufficiently high power of 2. We can certainly use a somewhat lower power of 2. Any higher power of 2 will suffice. THEOREM 1.3. For all k,n 1 and strictly dominating R [1,n] 2k , there exists A [1,n] k such that R[A\2 (8k)! -1] = A’. Furthermore, A [1,n] k is unique. We now incorporate the antichain concept. Let R [1,n] 2k . We say that A is an antichain for R if and only if A [1,n] k and A,RA are disjoint. Of course, we have the following familiar "maximal antichain" theorem.
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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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Pi01120305 - 1 P01 INCOMPLETENESS by Harvey M Friedman...

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