three quantifiers071402

three quantifiers071402 - 1 THREE QUANTIFIER SENTENCES by...

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1 THREE QUANTIFIER SENTENCES by Harvey M. Friedman* Ohio State University July 2, 2002 July 14, 2002 friedman@math.ohio-state.edu http://www.math.ohio-state.edu/~friedman/ Abstract. We give a complete proof that all 3 quantifier sentences in the primitive notation of set theory ( ,=), are decided in ZFC, and in fact in a weak fragment of ZF without the power set axiom. We obtain information concerning witnesses of 2 quantifier formulas with one free variable. There is a 5 quantifier sentence that is not decided in ZFC (see [Fr02]). 1. Preliminaries. Daniel Gogol [Go79] presents an argument that all 3 quantifier sentences in the primitive notation of set theory ( ,=) are decided in ZFC. In the author’s words, not all of the details are presented: “It is tedious but involves no difficulty to verify that if. ..”. p. 5, line 14. “This can be verified by considering all the possible cases, but is quite clear if considered carefully. So we omit what would be a very long verification.” p.8, end. We give a complete proof that all 3 quantifier sentences in set theory based on ,=, are decided in ZFC. In fact, we show that all sentences of somewhat higher complexity are decided in a weak fragment T of ZF without the power set axiom. We also give some strong information about witnessing 3 quantifier sentences that begin with an existential quantifier. Our main results are summarized in Theorem 11.1. We do not use ideas from [Go79].
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2 We work entirely within the primitive language of set theory, which is standard predicate calculus with equality and , using the standard quantifiers " , $ , and the standard connectives , , , Æ , . We assume the usual complete axioms and rules of classical logic. We let T be the following weak set theory. 1. Extensionality. 2. Pairing. 3. Union. 4. Infinity. 5. Foundation. 6. Bounded Separation. Pairing asserts the existence of {a,b}. Union asserts the existence of a. The usual formulation of Infinity in ZF is the existence of a set containing and closed under the operation that sends x to x {x}. Because we are only using Bounded Separation, we use the stronger version of infinity that asserts the existence of a set containing and closed under the operation that sends x,y to x {y}. Foundation asserts that every nonempty set has an epsilon minimal element. Bounded Separation asserts the existence of {x a: j }, where j is a formula in which all quantifiers are bounded. I.e., all quantifiers are of the forms ( " u v), ( $ u v), where u,v are distinct variables. In T, we can prove the existence of a least nonempty transitive set closed under power set, which we write as V( w ). In T, we can develop all of the basic facts about V( w ) and its elements, as well as define subsets of V( w ) and functions on V( w ) by recursion. Since the focus of the paper is on 3 quantifier sentences, we strictly follow the simplifying convention that all formulas will use at most the three distinct variables x,y,z. Letters such as u,v,w are used as metavariables over the official variables x,y,z, or variables used to carry out proof sketches that are to take place in T.
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three quantifiers071402 - 1 THREE QUANTIFIER SENTENCES by...

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