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EBRT090302

# EBRT090302 - 1 EQUATIONAL BOOLEAN RELATION THEORY by Harvey...

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1 EQUATIONAL BOOLEAN RELATION THEORY by Harvey M. Friedman Ohio State University [email protected] http://www.math.ohio-state.edu/~friedman/ 9/03/02 NOTE: THIS IS ONLY THE FIRST THREE SECTIONS, WHICH CONTAINS THE PROOF FROM LARGE CARDINALS. Abstract. Equational Boolean Relation Theory concerns the Boolean equations between sets and their forward images under multivariate functions. We study a particular instance of equational BRT involving two multivariate functions on the natural numbers and three infinite sets of natural numbers. We prove this instance from certain large cardinal axioms going far beyond the usual axioms of mathematics as formalized by ZFC. We show that this particular instance cannot be proved in ZFC, even with the addition of slightly weaker large cardinal axioms, assuming the latter are consistent. 1. EQUATIONAL BOOLEAN RELATION THEORY. Equational Boolean Relation Theory (equational BRT) concerns the Boolean equations between sets and their images under multivariate functions. We formally present equational BRT in this section. To be fully rigorous, we need to distinguish, say, a function of two variables from A into A and a function of one variable from A 2 into A. For this reason, we use the following definition of multivariate function. A multivariate function is a pair f = (g,k), where k 1 (the arity of f) and dom(g) is a set of ordered k-tuples. We put no other restrictions on g. We define dom(f) to be dom(g), and write f(x 1 ,...,x k ) = g(<x 1 ,...,x k >), where <x 1 ,...,x k > is the ordered k-tuple with coordinates shown. In practice, we need not be so careful about multivariate functions.

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2 Equational BRT is based on the following crucial notion of forward image. Let f be a multivariate function and A be a set. We define fA = {f(x 1 ,...,x k ): k is the arity of f and x 1 ,...,x k A}. We could write f[A k ] for this forward image construction, but it is particularly convenient to suppress the arity and write fA. In this way, f defines a special kind of operator from sets to sets. A BRT setting is a pair (V,K), where V is a set of multivariate functions and K is a set of sets. Typically, V and K are naturally related so that one is interested in the forward images of elements of V on elements of K, although in equational BRT, no restrictions are placed on the choice of V,K. We use N for the set of all nonnegative integers. We say that f is a multivariate function from A into B if and only if f is a multivariate function with dom(f) = A k and rng(f) B, where the arity of f is k. We use MF(A,B) for the set of all multivariate functions from A into B, and MF(A) for the set of all multivariate functions from A into A. We use S(A) for the set of all subsets of A, and INF(A) for the set of all infinite subsets of A. We say that f MF(N) is strictly dominating if and only if for all x dom(f), f(x) > max(x).
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