BabyBRT100301

BabyBRT100301 - 1 LECTURE NOTES ON BABY BOOLEAN RELATION...

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1 LECTURE NOTES ON BABY BOOLEAN RELATION THEORY* Harvey M. Friedman Department of Mathematics Ohio State University friedman@math.ohio-state.edu December 4, 2000 October 3, 2001 Abstract. This is an introduction to the most primitive form of the new Boolean relation theory, where we work with only one function and one set. We give eight complete classifications. The thin set theorem (along with a slight variant), and the complementation theorem are the only substantial cases that arise in these classifications. INTRODUCTION. Let f be a multivariate function of arity k. Let A be a set. We write fA for f[A k ]. Z = set of all integers, N = set of all nonnegative integers. THEOREM. Let f be a multivariate function from Z into Z. There exists an infinite A Z such that ????. QUIZ: Find four common mathematical symbols so that this is true and highly nontrivial. Here is the answer to the quiz. THIN SET THEOREM. Let f be a multivariate function from Z into Z. There exists an infinite A Z such that fA ≠ Z. DIGRESSION: TST is provable in ACA 0 ’ = RCA 0 + "for all x,n, the n-th jump of x exists". We have shown that TST is not provable in ACA 0 . See [FS00] or the proof below, which is better. It is open whether RCA 0 proves TST ACA 0 ’, or even whether RCA 0 proves TST Æ ACA 0 . RCA 0 cannot prove TST ACA 0 since ACA 0 is not finitely axiomatizable. We have also shown that TST for k = 2 is not provable in WKL 0 . See [CGH00]. Proof of TST: Use RT = Ramsey’s theorem. Wlog, replace Z by N. Let p be the number of order types of k-tuples. Set H(x) = f(x) if f(x) £ p; p+1 otherwise. Let A N be infinite and H- homogeneous in the sense that the value of H at tuples from A
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2 depend only on their order types. Then the values of f at tuples of any given order type from A are either all the same number £ p, or all > p. In any event, at least one of the numbers {0,. ..,p} is omitted by f on A. QED We can immediately ask what else can be put there other than the Boolean inequation fA ≠ Z. Think of Z as the universal set. This is what BRT = Boolean relation theory is all about. In these lecture notes, we only consider “baby” BRT, where we have just one function and one set. 1. SOME BOOLEAN ALGEBRA. The notion of Boolean algebra is a very robust concept with several different definitions. We give a particularly elegant definition. By way of motivation, the clearest examples of Boolean algebras are the Boolean fields of sets. These are structures (W, ,S, , « ,’), where W is a family of subsets of the set S closed under pairwise union, pairwise intersection, and complement relative to S. Here ’ is complementation relative to S. We let 2 be the structure ({0,1},0,1,+,•,-), where x+y = min(1,x+y), x•y is multiplication, and -x = 1-x. A Boolean algebra is a structure B = (B,0,1,+, ,-), where 0,1 B, +,•:B2 Æ B, -:B Æ B, such that every equation that holds universally in 2 also holds universally in B.
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BabyBRT100301 - 1 LECTURE NOTES ON BABY BOOLEAN RELATION...

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