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Unformatted text preview: 1 A COMPLETE THEORY OF EVERYTHING: validity in the universal domain by Harvey M. Friedman Department of Mathematics The Ohio State University May 16, 1999 Extended Abstract firstname.lastname@example.org www.math.ohio-state.edu/~friedman/ Introduction. A. Notions of predication on W. B. Basic formal theories of predication on W. C. Predication on W in class theory. D. Completeness of PC, PC(=), and PC(=,inf). E. Existential and universal sentences. 1. Principle of symmetric arguments. 2. Completeness. 3. Alternative axiomatizations. 4. Alternative determinations. F. and sentences. G. Arbitrary sentences. 1. Introduction. 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 usual interpretation of LPC(=) is the set interpretation, where domains are taken to be nonempty sets and relations and functions are taken to be set theoretic objects. The famous Gdel completeness theorem determines the formulas that are true in all set interpretations; i.e., the set validities. They are the formulas provable in PC(=). Here we consider a number of alternative interpretations of LPC(=), and discuss the corresponding validities. Under the domain interpretation of LPC(=), the domain is allowed to be any nonempty domain, regardless of whether that domain constitutes a set. For instance, V = the class of all sets, is a nonempty domain, but not a nonempty set. And W = 2 the domain consisting of everything, is even bigger, and certainly not a nonempty set. At the time of Frege, any distinction between domains and sets - or even between classes and sets - had not been made clear. For that matter, even with regard to the set interpretation of LPC(=), we have not presently made clear whether we are talking about pure sets in the sense of set theory (the cumulative hierarchy of sets), or sets with possible urelements as elements. Gdels determination of the set validities and domain validities is so robust as to be insensitive to such distinctions. We always arrive at the formulas provable in PC(=). Why is this so convincing? Because Gdels completeness theorem uses only a very small collection of assumptions (facts) about these concepts. Informally, to prove that every validity is derivable from simple axioms and rules, Gdel uses only the following: that there is a set or domain N of natural numbers and an appropriate successor operation, so that induction and recursion in a particularly elementary form can be performed over N with respect to relations and functions on N, which are used to create sets, relations, and functions on N and reason about them....
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