# FOM: What is the standard model for PA?

Mon Mar 23 17:11:15 EST 1998

```Torkel Franzen wrote:
>
>
>    >I think that after realizing that the natural numbers *may be
>    >seen* as constituting a very indeterminate "set" it is difficult
>    >to return to older, I would say, oversimplified picture as if
>    >nothing was happened. At least this is my case.  Probably you
>    >are able to be so "solid" in your opinion to not change your
>    >belief (or what it is) in standard model and simultaneously to
>    >realize possible vagueness of the same(?) model.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>   Well, I would say that considerations regarding feasibility inevitably
> lead to vague concepts (that may yet be mathematically and philosophically
> interesting and useful), but that this doesn't mean that the idealized
> version of the natural numbers - i.e. the natural numbers as ordinarily
> understood - is unclear or indeterminate.

What about "vagueness of the same(?) model"?

It is normal that an intuitive concept is vague. However, our
*mathematical* formalization of this concept should be possibly
as rigorous and determinate as e.g. the formalization by Peano
Arithmetic of even more vague concept of "all" natural numbers
implicitly involving the concept of feasible numbers.

>   >By the way, do you see now "indeterminateness of arbitrary
>   >property" of natural numbers in rather short segment
>   >0,1,2,...,1000 of natural numbers as in the case of "all"
>   >numbers or this set is still completely determinate for you?
>
>   Already the notion of "arbitrary property of 0" is indeterminate.
> The set 0,1,2....1000 is determinate, though, as is the set
> 0,1,2,... of all natural numbers. I'm not prepared to defend the
> notion of "arbitrary subset of the natural numbers" as determinate.

Let me formulate this question more definitely.  Do you see now
indeterminateness of the powerset of {1,2,...,1000} or,
alternatively, of the set 2^1000={0,1}^1000 of all finite
binary strings of the length 1000?

Actually this is rather unclear point. I will mainly present my
very informal *feeling* on what may happen here.  Let us put
aside Peano Arithmetic (PA) which neglects completely
feasibility and physical realizability concepts.  In particular
we cannot rely on the evidently infeasible process of creating
elements of the above set in lexicographical order. It is clear
that only some strings of this "set" exist(ed) or will be realized
in the (extremely indeterminate) future in our material world.
Anyway, I am not sure that any our formal theory, even PA, can
fix completely this set {0,1}^1000. It seems that there is some
analogy with the case of continuum (2^N, the powerset of
N={1,2,...}) which proves to be not fixed by ZFC due to G"odel
and Cohen results.  Is it "true" that

(A) the "set" of "simple" such strings (i.e.  those constructed
by a simple algorithm) like 00...0 (only zeros), 11...1 (only
ones), 0101...01 (alternating zeros and ones), etc.  exhaust
"all" strings of the length 1000

or

(B) we need to use inevitably a coin?

Does any abstract concept of random choice really fix the above
set of strings?

Note, that the above alternative (A) or (B) can be made more
precise as follows: Are "all" binary strings of the length 1000
generated by some fixed *feasibly computable* function

f:{1}^* -> {0,1}^1000?

Here {1}^* denotes the "set" of "all" finite unary strings of
*feasible* length. Put other way, is this set "enough" for our
hypothetical theory of feasibly finite binary strings (as it was
the case with G"odel constructible sets in ZFC) or we need
inevitably "non-feasibly-constructive" or "random" strings?

Only after (and simultaneously with) hard work on and with
formalization of feasibility we could get proper understanding
of this concept and related questions such as above. .

>   >Our understanding and "justification" goes *in terms*, and via
>   >numerous repetitions of using some formal rules. Otherwise, how
>   >to teach children to mathematics?
>
>   How we actually learn arithmetic is a difficult question. I don't
> find any ideas to the effect that it *must* happen in some particular
> way convincing. Certainly rules play a large role, but what is it to
> learn a rule, and what conditions must be satisfied if we are to
> be able to learn a rule?

I agree that these questions are indeed difficult. Actually, I
had intention to say by the words "goes *in terms*" and "via"
that formal rules and informal understanding cannot be separated
one from another and their roles are at least equal. I.e., NOT
"first informal understanding of a concept, and only then
creation and justification of related rules". Therefore the role
of formal rules -- of our *subjective* but of course not
completely free creations -- is at least higher than it is
sometimes considered in discussions on f.o.m.