BRT,CMU032909

BRT,CMU032909 - BOOLEAN RELATION THEORY by Harvey M....

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Unformatted text preview: BOOLEAN RELATION THEORY by Harvey M. Friedman Ohio State University [email protected] http://www.math.ohio-state.edu/%7Efriedman/ Mathematical Logic Seminar Carnegie Mellon University March 27, 2009 BABY BRT BRT is always based on a choice of BRT setting. A BRT setting is a pair (V,K), where V is an interesting family of multivariate functions. K is an interesting family of sets. In this talk, we will only consider V,K, where V is an interesting family of multivariate functions from N into N. K is an interesting family of subsets of N. Here N is the set of all nonnegative integers. BRT is always based on the following dimension suppressing forward imaging operator. Let f be a k-ary function. I.e., all elements of dom(f) are k- tuples. Let A be a set. f A = f[A k ] = {f(x 1 ,...,x k ): x 1 ,...,x k ∈ A}. BABY BRT There are two flavors of Baby BRT. Equational BRT. Inequational BRT. In Equational BRT, we focus on all statements of the following form: FOR ALL f ∈ V, THERE EXISTS A ∈ K, SUCH THAT A GIVEN BOOLEAN EQUATION HOLDS BETWEEN A,fA. In Inequational BRT, we focus on all statements of the following form: FOR ALL f ∈ V, THERE EXISTS A ∈ K, SUCH THAT A GIVEN BOOLEAN INEQUATION HOLDS BETWEEN A,fA. Here we use N as the Universal Set for Boolean algebra purposes. BABY BRT We now give the two seminal examples of Equational and Inequational Baby BRT. For the example of Inequational Baby BRT, we use V = MF, K = INF, where MF is the family of all functions f whose domain is some N k and whose range is a subset of N. INF is the family of all infinite subsets of N. THIN SET THEOREM. ( ∀ f ∈ MF)( ∃ A ∈ INF)(fA ≠ N). For the example of Equational Baby BRT, we use V = SD, K = INF, where SD is the family of strictly dominating f ∈ MF, in the sense that for all x 1 ,...,x k ∈ N, f(x 1 ,...,x k ) > max(x 1 ,...,x k ). COMPLEMENTATION THEOREM. ( ∀ f ∈ SD)( ∃ A ∈ K)(fA = N\A). THIN SET THEOREM THIN SET THEOREM. ( ∀ f ∈ MF)( ∃ A ∈ INF)(fA ≠ N). Proof: Let f:N k → N. Let p be the number of order types of k-tuples from N. By the infinite Ramsey theorem, we can find infinite A such that f assumes at most one value in {0,...,p} when using arguments from a single order type. Hence f omits at least one value from {0,...,p}. QED We know that TST is provable in ACA’ but not in ACA . Also TST for k = 2 is not provable in WKL . These results of ours are proved in Peter Cholak, Mariagnese Giusto, Jeffry Hirst, and Carl Jockusch, Free sets and reverse mathematics, in: Reverse Mathematics, ed. S. Simpson, Lecture Notes in Logic, Association for Symbolic Logic, 1905. http://www.nd.edu/~cholak/papers/preincollection.html H. Friedman and S. Simpson, Issues and problems in reverse mathematics, 127-144, in: Computability Theory and Its Applications, ed. Cholak, Lempp, Lerman, Shore, American Mathematical Society, 2000....
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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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BRT,CMU032909 - BOOLEAN RELATION THEORY by Harvey M....

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