Pi01120405 - 1 P01 INCOMPLETENESS by Harvey M. Friedman...

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1 P 0 1 INCOMPLETENESS by Harvey M. Friedman December 4, 2005 ‘Beautiful’ is a word used by mathematicians with a semi rigorous meaning. We give an ‘arguably beautiful’ explicitly P 0 1 sentence independent of ZFC. See Proposition A(4,3) from section 1. 1. P 0 1 INDEPENDENT STATEMENT. 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. Here is a trivial modification of Theorem 1.1 with the same proof. We call this the ‘complementation theorem’ for R[A\{t} k ]. THEOREM 1.3. For all k,n,t 1 and strictly dominating R [1,n] 2k , there exists A [1,n] k such that R[A\{t} k ] = A’. Furthermore, A [1,n] k is unique.
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2 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. I.e., RA A’. The reader will recall the following familiar ‘maximal antichain’ theorem. THEOREM 1.4. For all k,n 1, every R [1,n] 2k has a maximal antichain. Note that the equation in Theorems 1.2, RA = A’, asserts that A is a very strong kind of antichain. We refer to the following trivial restatement of Theorem 1.2 as a ‘complete antichain’ theorem. THEOREM 1.5. For all k,n 1, every strictly dominating R [1,n] 2k has an antichain A such that RA = A’. Furthermore, A is unique. Note that we could equally well write RA A’ instead of RA = A’, since we are also asserting A is an antichain for R.
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Pi01120405 - 1 P01 INCOMPLETENESS by Harvey M. Friedman...

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