# FOM: What is the standard model for PA?

Thu Mar 26 16:39:30 EST 1998

```Martin Davis wrote:
>
> At 06:40 PM 3/25/98 +0300, sazonov at logic.botik.ru wrote:
>
> >In the light of discussion on feasible numbers (and also in a
> >general context) I have a question to everybody.
> >
> >What makes the powerset 2^N of natural numbers (i.e. the set of
> >infinite binary strings) to be indeterminate *in contrast to*
> >the powerset 2^1000={0,1}^1000 of {1,2,...1000} which should be
> >determinate (according to the traditional view and *contrary* to
> >my intuition)?  What is the crucial difference? Is this
> >difference based on *infinity* of N and in *finity* of
> >{1,2,...1000}? Note, that 2^1000 (as well as 2^N) behaves now
> >*as an infinity* from the point of view of feasibility. I
> >believe that comparison would be interesting.
> >
>
> Indeed a good question. My answer would be that both are perfectly determinate.

Strange. It seems I remember that you expressed the opinion
that the universe of sets is indeterminate in contrast to the
natural numbers, or something like this.

Anyway, I hardly could say anything more original on 2^N that it is
usually considered by the corresponding specialists in f.o.m. First,
what is an arbitrary property of natural numbers? We know from
paradoxes that this concept is somewhat dangerous. The well-known
results of G"odel and Cohen witness that this notion is
essentially indeterminate. The term "property" assumes that it is
expressed in a language with an appropriate semantics. We have only
some examples of such languages. Then the questions of
(im)predicativity arise. May be highest professionals like Professor
Feferman would make clearer what is happened here. As to me, I see
no reason to consider 2^N as determinate or even to use
auto-suggestion of complete determinateness with the hope to get
anything useful.

Now, what is the difference with the (infeasible!) powerset 2^1000?
It seems we can equally apply the above considerations (or doubts)
like those on (im)predicativity to this case (except of the direct
references to G"odel and Cohen). Am I right?