EquatReps092310

# EquatReps092310 - EQUATIONAL REPRESENTATIONS by Harvey M...

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EQUATIONAL REPRESENTATIONS by Harvey M. Friedman* Ohio State University http://www.math.ohio-state.edu/%7Efriedman/ September 23, 2010 NOTE: THIS IS AN ADVANCED DRAFT THAT NEEDS SOME POLISHING BEFORE SUBMISSION FOR PUBLICATION. 1. Preliminaries: N, N, ℘℘ N. 2. Fundamental Lemma Concerning Clones. 3. From ( f1,...,fk)( x1,...,xp)( ϕ ) To ( binary f)( x,y)( ϕ ). 4. From N, N, ℘℘ N to N N, ( N N). 5. From N N, ( N N) To Binary Sentences Over ( N N,...). 6. From Binary Sentences Over ( N N,...) To Binary Sentences Over ( N, ). 7. From Binary Sentences Over ( N, ) To Binary Sentences Over ( ,<). 8. From Binary Sentences Over ( ,<) To ( f: )( x,y )(s = t). 1. Preliminaries: N, N, ℘℘ N. We begin by presenting the language L(N, N, ℘℘ N). This is the standard language for presenting third order sentences, using its intended interpretation. There are variables n i , i 1, over N. Variables x i , i 1, over N. Variables A i , i 1, over ℘℘ N. The terms are defined inductively by Each n i is a term. 0 is a term. If s,t are terms, then S(t), s+t, s t are terms. The atomic formulas are x i = x j . A i = A j . x i A j . t x i , where t is a term. s = t, where s,t are terms.

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Formulas of L(N, N, ℘℘ N) are obtained using connectives and quantifiers in the usual way. Their intended meaning is obvious. The Σ 1 formulas of L(N, N, ℘℘ N) are the formulas without any quantifiers over ℘℘ N. The Σ 2 1 formulas of L(N, N, ℘℘ N) are the formulas of L(N, N, ℘℘ N) are the formulas that begin with zero or more existential quantifiers over ℘℘ N, followed by a Σ 1 formula of L(N, N, ℘℘ N). We use ZC for Zermelo set theory with the axiom of choice, which we will take to be ZFC with Replacement omitted. The Σ 21 sentences of L(N, N, ℘℘ N) are well known to be very expressive. For example, the continuum hypothesis is provably equivalent to a Σ 2 1 sentence, over ZC. Throughout the paper, we take the Σ 2 1 sentences to be the Σ 2 1 sentences of L(N, N, ℘℘ N). Here is the main result of this paper. THEOREM 7.5. The continuum hypothesis (more generally, every Σ 21 sentence) is provably equivalent, over ZC, to a sentence of the form ( f: )( x,y )(s = t) where s,t are terms in f,x,y,+, ,0,1. Here are some other results of significance. THEOREM 6.1. The continuum hypothesis (more generally, every Σ 2 1 sentence) is provably equivalent, over ZC, to a sentence of the form ( f: 2 )( x,y )( ϕ ) where ϕ is quantifier free in f,x,y,<. THEOREM 5.1. The continuum hypothesis (more generally, every Σ 21 sentence) is provably equivalent, over ZC, to a sentence of the form ( f:( N) 2 N)( x,y N)( ϕ )
where ϕ is quantifier free in f,x,y, .

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