[FOM] strong hypotheses and the theory of N

[Ali Enayat]

In a recent posting (March 15, 2010), Joe Shipman has asked:

>A related question: is there a natural way to represent the
>"arithmetical content" of ZF by arithmetical axioms; in other words, a
>natural decidable set of arithmetical statements which have the same
>arithmetical consequences as ZF?

The answer is positive.

The arithmetical content of any recursive extension ZF* [in particular
ZF, or ZF + large cardinals] is axiomatized by arithmetical sentences
of the following form (each is an implication)

S --> (Con (S* + T)),

where:

S ranges over sentences of the language of arithmetic;
S* is the sentence in the language of set theory the expresses "S
holds in (omega, plus, times)" ;
T ranges over finite subsets of the ZF*-axioms; and
Con(X) is the arithmetical sentence asserting the formal consistency
of the theory X.

Indeed, the result is more general and can be easily adapted to other
"reflexive theories" besides ZF (those proving the formal consistency
of each of their finite subtheories) such as second order arithmetic,
and Kelley-Morse set theory.

The above result is included in a forthcoming paper of mine that
focuses on models of PA that arise as standard models of arithmetic in
models of ZF.

As shown in the paper, Analytic (and in particular countable or Borel)
nonstandard models of PA that arise as standard models of arithmetic
in some nonstandard model of ZF are precisely those nonstandard models
that satisfy the aforementioned recursive set of axioms (with ZF* =
ZF), and which are additionally (1) recursively saturated, and (2)
have countable cofinality.

Regards,

Ali Enayat