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UnprovThms052605

# UnprovThms052605 - 1 UNPROVABLE THEOREMS by Harvey M...

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1 UNPROVABLE THEOREMS by Harvey M. Friedman [email protected] http://www.math.ohio-state.edu/%7Efriedman/ Cal Tech Math Colloq April 19, 2005 INTRODUCTION. We discuss the growing list of examples of simply stated Theorems where it is known that there are no concrete proofs. Here is our agenda. A. In sufficiently long finite sequences from a finite set, certain blocks are subsequences of certain later blocks. B. In sufficiently tall finite trees, certain truncations are embedded in certain taller truncations. C. Multivariate functions on the integers have nonsurjective infinite restrictions. D. Any two countable sets of reals are pointwise continously comparable. E. Any permutation invariant Borel function from infinite sequences of reals into infinite sequences of reals maps some sequence into a subsequence. F. For every symmetric Borel set in the plane, either it or its complement has a Borel selection. G. Any Borel set in the plane that has a Borel selection on every compact set has a Borel selection. H. For any two multivariate functions on the natural numbers of expansive linear growth there are three infinite sets which bear a certain Boolean relation with their images under the two functions. (Boolean relation theory). 1. BLOCK SUBSEQUENCE THEOREM. The block subsequence theorem involves a single finite string in k letters. The following binary case is elementary, and a good challenge for gifted high school students. THEOREM 1.1. There is a longest finite sequence x 1 ,...,x n in two letters such that no consecutive block x i ,...,x 2i is a subsequence of a later consecutive block x j ,...,x 2j .

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2 The longest length is 11, with 12221111111 and 21112222222 as the only examples. THEOREM 1.2. There is a longest finite sequence x 1 ,...,x n in 3 letters where no consecutive block x i ,...,x 2i is a subsequence of a later consecutive block x j ,...,x 2j . Theorem 1.2 merely states the existence of a natural number with a specific testable property. But the simplest known way to prove this involves not only infinite sequences but also defining infinite sequences using all infinite sequences (impredicativity). This is a weak use of the uncountable. A logically more down to earth but more involved proof can be given for any number of letters. The most logically economical proof uses induction with an induction hypothesis that has three alternating quantifiers over the natural numbers. Two quantifiers does not suffice. Coming back to the case of three letters, the exotic nature of all proofs is illustrated by the following lower bound on the length of the longest such finite string. In the case of two letters it is 11. In the case of three letters, it is greater than the 7198-th Ackermann function at 158,386. The 3rd Ackermann function at 158,386 is already an exponential tower of 2’s of height 158,386.
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UnprovThms052605 - 1 UNPROVABLE THEOREMS by Harvey M...

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