[FOM] Strong Hypotheses and the Theory of N

[Robert Solovay]

In a recent posting (March 20th) Joe Shipman asks the following question:

Would the problem be easier if I asked for a (presumably consistent)
“natural” set of arithmetical axioms which implied all the arithmetical
consequences of ZF but was not necessarily limited to them?

This relates to his prior question (in a posting on March 15th):

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?

I plan to give answers to both these questions in this posting. My
answer to the second question is similar to that of Ali Enayat (though
our two answers were independently developed).

I need the notion of $n$-consistency. A theory (with a version of
arithmetic) is $n$-consistent if every Sigma^0_n sentence proved by
the theory is true.  A key point is that, e. g., the statement "ZFC is
$n$-consistent" is arithmetical. This is because we can define, in
Peano Arithmetic, the notion of truth for Sigma^0_n sentences. (And we
can prove the appropriate instances of Tarski's ("snow is white" is
true iff snow is white) schema.

Then the answer to the first of Shipman's questions that I propose is:

1) The axioms of Peano Arithmetic;

2) For each positive integer $n$, the arithmetical formulation of "ZFC
is $n$-consistent".

That all these arithmetical formulas are true can be proved, e. g., in
ZFC + "There is an inaccessible cardinal".

My answer to the second question is similar. I need to define a
sequence of approximations to ZFC, T_n (for n in omega) that ZFC can
validate and that converge to ZFC. There are many ways to do this. For
example, we can take T_n to be all the axioms of ZC (Zermelo set
theory with choice and foundation) plus all the instances of Sigma_n
replacement.

The important fact is that ZFC can prove that each of the T_n's has an
omega model. This is easily proved using "reflection".

Then the theory which exactly captures the arithmetical consequences of ZFC is:

1) Peano Arithmetic;

2) For every pair of positive integers n and m an axiom asserting that
T_n is $m$-consistent.

By the way, it seems to me (though I cannot prove this) that Enayat
should have added to his list of axioms some weak theory (Robinson's Q
would suffice) which can prove all true Sigma^0_1 sentences.
Otherwise, I do not see how to prove his theory has the stated
properties.

--Bob Solovay