Pi01*113005 - 1 This is an MS Word formatted version of my...

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1 This is an MS Word formatted version of my FOM posting #258 of November 27, 2005, at http://www.cs.nyu.edu/pipermail/fom/2005- November/009395.html Quite a number of typographical errors in the FOM posting have been corrected. P 0 1 INCOMPLETENESS by Harvey M. Friedman November 30, 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 Proposition B from section 2, and variants. Proposition A is simplest and has a graph theoretic flavor, with algebraic overtones. Proposition B has a fixed point flavor. 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 the basic ‘complementation theorem for RA’. THEOREM 1.1. 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
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2 coordinate of x} = "A with t omitted". Here is a modification of Theorem 1.1 which we call the ‘complementation theorem for R[A\(8k)!]’. THEOREM 1.2. For all k,n 1 and strictly dominating R [1,n] 2k , there exists A [1,n] k such that R[A\(8k)!] = 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 "maximal antichain" theorem. THEOREM 1.3. For all k,n 1, every R [1,n] 2k has a maximal antichain. Note that Theorem 1.3 is virtually contentless, since we are in a finite context where every nonempty class has a maximal element. Note that Theorem 1.1 provides a much stronger kind of antichain, which we call a ‘complete’ antichain. We refer to the following as a ‘complete antichain’ theorem. THEOREM 1.4. For all k,n 1, every strictly dominating R [1,n] 2k has an antichain A such that RA contains A’. Furthermore, A is unique. However, the analog of Theorem 1.4 for R[A\(8k)!] is false. THEOREM 1.5. The following is false. For all k,n 1, every strictly dominating R [1,n] 2k has an antichain A such that R[A\(8k)!] contains A’. We will weaken the conclusion in a simple way. Our development depends heavily on a very strong regularity condition on R. We say that R [1,n] k is order invariant if and only if for all x,y in [1,n] k of the same order type, R(x) iff R(y). The number of such R is bounded by an exponential expression in k that does not depend on n.
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3 The imposition of order invariance is still not sufficient: THEOREM 1.6. The following is false. For all k,n 1, every strictly dominating order invariant R [1,n] 2k has an antichain A such that R[A\(8k)!] contains A’.
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Pi01*113005 - 1 This is an MS Word formatted version of my...

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