1.4ThinSet071110

# 1.4ThinSet071110 - 1 1.4 Thin Set Theorems Recall the Thin...

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1 1.4. Thin Set Theorems. Recall the Thin Set Theorem from section 1.1. THIN SET THEOREM. For all f MF there exists A INF such that fA N. The Thin Set Theorem as written above is a statement of IBRT in A,fA on (MF,INF). This specific statement is due to the present author, who studied it for its significance for Reverse Mathematics and recursion theory. A variant of this statement was already introduced much earlier in the literature on the (square bracket) partition calculus in combinatorial set theory, in [EHR65]. In their language, the Thin Set Theorem reads ( n < ω )( ω [ ω ] n ω ). The n indicates a coloring of the unordered n-tuples from the ω to the left of , the lower ω indicates the number of colors, and the ω in [ ] indicates the cardinality of the “homogenous” set. But here [ ] indicates a weak form of homogeneity - that at least one color is omitted. The mathematical difference between this square bracket partition relation statement and the Thin Set Theorem is that the former involves unordered tuples, whereas the latter involves ordered tuples. However, see Theorem 1.4.2 below for an equivalence proof in RCA 0 . Also see [EHMR84], Theorem 54.1. It was immediately recognized that this square bracket partition relation follows from the usual infinite Ramsey theorem, which is written in terms of the round parenthesis partition relation ( n,m < ω )( ω ( ω ) n m ). Experience reveals that when the Thin Set Theorem is stated exactly in our formulation above (with ordered n tuples), mathematicians who are not experts in the partition calculus, do not recognize the Thin Set Theorem’s connection with the partition calculus and combinatorial set theory. They are struck by its fundamental character, and will not be able to prove it in short order. They apparently would have to rediscover the infinite Ramsey theorem, and in our experience, long before they invest that kind of effort, they demand a proof from us.

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2 The Thin Set Theorem - as an object of study in the foundations of mathematics - first appeared publicly in [Fr00], and in print in [FS00], p. 139. There we remark that it trivially follows from the following well known Free Set Theorem for N. FREE SET THEOREM. Let k 1 and f:N k N. There exists infinite A N such that for all x A k , f(x 1 ,...,x k ) A f(x 1 ,...,x k ) {x 1 ,...,x k }. The implication is merely the observation that if A obeys the conclusion of the Free Set Theorem, then A\{min(A)} obeys the conclusion of the Thin Set Theorem (min(A) is not a value of f on (A\{min(A)}) k ). The Free Set Theorem is easily obtained from the infinite Ramsey theorem in a well known way. Choose infinite A N such that the truth value of f(x 1 ,...,x k ) = y depends only on the order type of x 1 ,...,x k ,y, provided x 1 ,...,x k ,y A. If f(x
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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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1.4ThinSet071110 - 1 1.4 Thin Set Theorems Recall the Thin...

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