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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 02
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.
11CA
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 1consistency of predicative analysis”,
1982, 197230 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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View Full Documentconsidered 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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 Fall '08
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 Math

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