If an arithmetical statement is proved using the Axiom of Determinacy (AD), what must one believe about AD...

In article <TORKEL.89Oct20212507@tyfon.sics.se> torkel@sics.se (Torkel
Franzen) writes:

>... So, to make this clearer: an arithmetical
>statement might be proved using AD, which to my mind would not at all
>serve to show that the statement is true, or (say) using the existence
>of a _weakly_ compact cardinal, which I would regard as establishing
>the truth of the statement. I introduce the example only to point to a
>distinction or question that falls away on a formalistic view, not to
>put forward my particular views of what is convincing.

Here's a technical note which has some bearing on the philosophical
discussion.  Suppose one has proven an arithmetical statement using AD.
To be convinced that the arithmetical statement is true, you don't have
to believe that AD is true, only that AD is omega-consistent.  Saying
that AD is omega-consistent is not only much weaker than saying that AD
is true, it is a very different sort of statement.  Saying that AD is
omega-consistent is basically not an ontological claim about the
mathematical universe, while a claim that AD is true clearly is such an
ontological claim.  (There's nothing very special about AD in this
discussion; you can replace it with a large cardinal axiom if you
prefer.)

And casual observation indicates that most set theorists believe that
ZF + AD is omega-consistent (even that it has a well-founded model),
while most do not believe that AD is true (some of these take the
tolerant attitude described earlier in this thread and deny the
meaningfulness of the question, and some simply believe that AD is
false).  I mention this to illustrate the fact that the
omega-consistency of a theory is much easier to believe in than the
actual truth of the theory.

(By the way, some of the people who believe that AD is simply false
have done substantial work on AD; often the justification given is that
they believe that AD is true in L[R], an important inner model of the
universe, even though they believe that the axiom of choice is true in
the mathematical universe itself.)

-- Mitchell Spector <spector%sumax.uucp@beaver.cs.washington.edu>

In a follow-up, Spector wrote:

Torkel Franzen objected that omega-consistency wasn't strong enough to
make this claim, and I think he's right.  When I used the phrase "AD is
omega-consistent", what I meant was "AD has an omega-model" (or,
equivalently, ZF + AD is consistent in omega-logic).  Subject to this
change, I stand by everything I said.

In general, it's possible for a theory to be omega-consistent but to
have no omega-model.  For example, add to the language of arithmetic
aleph-one many new constant symbols, and take the theory T to be
first-order Peano arithmetic together with all sentences of the form
"c <> d", where c and d are distinct new constant symbols.  Clearly T
has no omega-model.  But T is omega-consistent (a proof appears below).
Does anybody know of an example in a countable language?  Or, since
this is what we were discussing, an example which is an extension of ZF
in the language of set theory?

Here's a proof of the omega-consistency of T above.  Let T prove a
sentence of the form (there exists x)(phi (x)).  Then there exist
finitely many of the new constant symbols c1, ..., cn, such that
PA + "c1, ..., cn are pairwise distinct" proves that sentence and such
that all the new constant symbols appearing in phi are among
c1, ..., cn.  Writing phi (x) as phi (x, c1, ..., cn), it follows that
PA proves the following sentence:

  (for all y1, ..., yn) (if y1, ..., yn are pairwise distinct, then
  (there exists x) (phi (x, y1, ..., yn))).

It follows that there is a natural number k such that, in omega,
phi (k, 1, 2, ..., n) holds.  Let A be any elementary extension of
omega of cardinality aleph-one.  Expand A to a model of our language by
defining the interpretations of c1, c2, ..., cn in A to be 1, 2, ..., n
(in A), respectively, and define the interpretations of the other new
constant symbols arbitrarily, just making sure that distinct constant
symbols are interpreted as distinct elements of A.  (This can be done
since A is uncountable.)  It follows that A is a model of
T union {phi (k)}.  So there is a (standard) natural number k such that
T is consistent with phi (k), as desired.

-- Mitchell Spector <spector%sumax.uucp@beaver.cs.washington.edu>

[ Editor: no answer to the question about a countable language was
  found in the collected postings. ]