BoolRelnThyNotes100601

BoolRelnThyNotes100601 - 1 BOOLEAN RELATION THEORY NOTES by...

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1 BOOLEAN RELATION THEORY NOTES by Harvey M. Friedman* Ohio State University [email protected] http://www.math.ohio-state.edu/~friedman/ February 15, 2001 October 6, 2001 Abstract. We give a detailed extended abstract reflecting what we know about Boolean relation theory. We follow this by a proof sketch of the main instances of Boolean relation theory, from Mahlo cardinals of finite order, starting at section 19. The proof sketch has been used in lectures. 1. MULTIVARIATE FUNCTIONS. Boolean relation theory (BRT) concerns the relationship between sets and their images under multivariate functions. To avoid ambiguities, we officially define a multivariate function to be a pair f = (g,k), where k 1 (the arity of f) and dom(g) is a set of ordered k-tuples. We take dom(f) to be dom(g), and write f(x 1 ,...,x k ) = g(<x 1 ,...,x k >). In practice, we will not be so careful about multivariate functions. BRT is based on the following CRUCIAL notion of forward image. Let f be a multivariate function and A be a set. fA = {f(x 1 ,...,x k ): k is the arity of f and x 1 ,...,x k A}. We could have written f[A k ], but want to suppress the arity and write fA. In this way, f defines a special kind of operator from sets to sets. 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) Õ B, where f has arity k. 2. EQUATIONAL, INEQUATIONAL, PROPOSITIONAL, BOOLEAN RELATION THEORY.
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2 In BRT, we will use set variables A 1 ,A 2 ,..., and function variables f 1 ,f 2 ,... . The BRT atoms consist of ,U,A i , and f i A j , for any i,j 1. BRT expressions are defined as the least set containing the BRT atoms, where if s,t are BRT expressions then so are s t, s « t, s’. The idea here is that U is the universal set, and s’ is U\s. A BRT equation is s = t, where s,t are BRT expressions. A Boolean inequation is s t, where s,t are BRT expressions. Let V be a set of multivariate functions and K be a set of sets. Let n,m 1. Equational (inequational, propositional) BRT on (V,K) of type (n,m) analyzes statements of the form: for all f 1 ,...,f n V, there exists A 1 ,...,A m K, such that a given BRT equation (BRT inequation, propositional combination of BRT equations) holds among the sets and their images under the functions. More formally, for all f 1 ,...,f n V, there exists A 1 ,...,A m K, such that a given BRT equation (BRT inequation, propositional combination of BRT equations) holds, where we require that in the given BRT equation (BRT inequation, propositional combination of BRT equations), all BRT atoms that appear are among the m(n+1) BRT atoms A 1 ,...,A m f 1 A 1 ,...,f 1 A m ... f n A 1 ,...,f n A m . For this purpose, we take the universal set U to be the union of the ranges of the elements of V and the elements of K. Most typically, this will simply be the largest element of K (which is not guaranteed to exist).
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3 BRT of type (1,1) is called unary BRT. Here we just have one function and one set. Even equational BRT on (V,K) of type (1,1) can be deep. See section 18 below, and [Fr01], which scratches the surface of unary BRT.
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