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1 A COMPLETE THEORY OF EVERYTHING: satisfiability in the universal domain Harvey M. Friedman October 10, 1999 [email protected] http://www.math.ohio-state.edu/~friedman/ 1. GENERAL REMARKS Let LPC(=) be the usual language of predicate calculus with equality, and PC(=) be a standard system of axioms and rules and inference for predicate calculus with equality. The completeness theorem determines those sentences that have models with certain domains. Here are five cases: 1. Nonempty sets. 2. N. 3. Infinite sets. 4. Fixed infinite set. 5. Extensions of unary predicates. Very few basic principles are needed for these results. 4 needs the axiom of choice. Note that 5 does not have the accepted clarity of meaning that 1-4 have. In fact, we will be considering two major distinct interpretations of 5. Nevertheless, the completeness theorem applies to 5, since its proof is so robust. We call 5 the domain interpretation. Here we take the view that LPC(=) is applicable to structures whose domain is too large to be a set. This is not just a matter of class theory versus set theory, although it can be interpreted as such, and this interpretation is discussed briefly at the end. We apply LPC(=) to the largest domain of all - the domain W consisting of absolutely everything. In order to make sense of this, we need the concept of arbitrary predicate of several variables on W.
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2 This context is often viewed as a dangerous slippery slope, fraught with paradoxes, or at least, quite murky, subject to differing interpretations. But the thrust of this work is that surprisingly minimal principles about predication on W are needed to establish which sentences of LPC(=) are satisfiable in domain W, and related results. 2. TWO NOTIONS OF PREDICATION AND INTERPRETATIONS In the context of predication on W, identity of indiscerni- bles states the following: for all x,y, x = y iff for all unary predicates P on W, P(x) P(y). This follows from singleton extensions principle: for all x there exists a unary predicate P on W such that " y(P(y) y = x). In mathematical and set theoretic predication, the singleton extensions principle holds. However, there is another concept of predication discussed by philosophers, for which the singleton extensions principle is doubtful. This is the notion of predicates that are given without reference to any particular objects, but only involving concepts. Or one can take a linguistic tack - relations that can be defined in a language, where language is broadly interpreted, rather than in some particular already formalized language. We will use the term "general predicate" for the first concept of predication, where the singleton extensions principle obviously holds. And we will use the term "pure predicate" for the second concept of predication, where even the doctrine of identity of indiscernibles is problematic.
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