Conservation9=24=99

Conservation9=24=99 - CONSERVATION by Harvey M. Friedman...

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CONSERVATION by Harvey M. Friedman September 24, 1999 John Burgess has specifically asked about whether one give a finitistic model theoretic proof of certain conservative extension results discussed in [Si99]. Burgess asks this in connection with Hilbert’s program. Specifically, I. WKL 0 is a conservative extension of PRA for -0-2 sentences. II. ACA 0 is a conservative extension of PA for arithmetic sentences. III. ATR 0 is a conservative extension of IR for arithmetic sentences. IV. -1-1-CA 0 is a conservative extension of ID(<omega) for arithmetic sentences. Here IR is Feferman’s IR, which can be taken to be the theory extending PA with new function symbols encoding the Kleene H- sets on each specific initial segment of the ordinal notation system Gamma0. All of these conservative extension results are given model theoretic proofs in [Si99] except for III. The reader is referred to the exposition of my proof in Friedman, MacAloon, Simpson, “A finite combinatorial principle which is equivalent to the 1-consistency of predicative analysis”, 1982, 197-230 in: G. Metakides (ed.), Patras Logic Symposion, Studies in Logic and the Foundations of mathematics, North- Holland, 1982. We write PFA (polynomial function arithmetic) for the system in the language of 0,1,+,x,<, with the usual successor axioms and defining equations, together with induction for all bounded formulas. This is the same as what is called I 0 in the book {HP93] and elsewhere. In [HP93], it is proved that PFA is fully capable of developing finite sequence coding and formalizing syntax. In fact, they devote all of Chapter V, section 3, to this topic, which is entitled “Exponentiation, Coding Sequences and Formalization of Syntax in I 0 .” In the Bibliographic Notes on page 406, they write “A formalization of syntax in I 0 is
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considered here for the first time, though the ideas on which it is based have been around for some time.” This makes PFA a good vehicle for taking a reverse mathematics point of view towards weak fragments of arithmetic. From this point of view, it is natural to take EFA = exponential function arithmetic to be in the language of PFA, whose axioms are PFA plus the single axiom (forall n)(2 n exists). And we take EFA* to be PFA plus the axiom that asserts that for all n, there is a sequence of integers of length n, starting with 0, where each term is the base 2 exponentiation of the preceding term. Here is what we will do in this note. 1. We isolate a crucial general fact about theories, which we call the Key Lemma. The Key Lemma has an easy model theoretic proof. But the Key Lemma also has a proof using the Criag interpolation theorem for predicate calculus with equality. The interpolation theorem has model theoretic proofs, but it also has a proof theoretic proof using Gentzen’s cut elimination theorem for predicate calculus with equality. The usual proof of the cut elimination theorem with iterated exponential estimates (given by Gentzen) is readily formalized in EFA’. This yields a proof of the interpolation
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Conservation9=24=99 - CONSERVATION by Harvey M. Friedman...

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