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