FOM: What is the standard model for PA?

Vladimir Sazonov sazonov at
Mon Mar 23 17:11:15 EST 1998

Torkel Franzen wrote:
>   Vladimir Sazonov says:
>    >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 


(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. 

Vladimir Sazonov
Program Systems Institute,  	| Tel. +7-08535-98945 (Inst.),
Russian Acad. of Sci.		| Fax. +7-08535-20566
Pereslavl-Zalessky,		| e-mail: sazonov at
152140, RUSSIA			|

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