FOM: What is the standard model for PA?

[Torkel Franzen]

  Vladimir Sazonov says:

  >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.

  I doubt that any canonical formalization of the notion of feasible
number will emerge, but this remains to be seen. As you are aware, I
don't think there is any objective justification for the view that the
concept of "all natural numbers" is vague or that it implicitly
involves the concept of feasible numbers. (That we learn to speak of
the totality of natural numbers by using feasible numbers does not
imply that the concept of feasible number is involved in the concept
of natural number.)

   >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?

  This is perfectly determinate, from the point of view of
ordinary mathematics (i.e. when one is speaking of the natural numbers
in the ordinary sense). However, as soon as one introduces the notion
of feasibility, indeterminacy inevitably appears. So I would say
that the powerset of {1,2,...,1000} is determinate, but "the
feasible powerset of {1,2,...1000}" is indeterminate.

  >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.

  I'm sure you're right on this score, if by "this concept and
related questions" you have in mind concepts related to feasibility.

  >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".

  The plausibility of this view would depend on what you count as a
"formal rule". The vast majority of those who are familiar with the
natural numbers and with elementary arithmetic know nothing of PA or
proofs by induction or first order logic.

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Torkel Franzen