FOM: What is the standard model for PA?

[Torkel Franzen]

  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.

  >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.

  >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?

  >Take it as axiom which has experimental confirmation.

  I would think, rather, that the commutativity of addition is evident
to people who wouldn't recognize an inductive proof if it bit them on
the nose because they use another, intuitive, argument to convince
themselves that addition is commutative. For example, maybe we see
it as evident that the outcome of a counting operation will be the
same in whatever order we count the objects. However, I'm uncertain
about how to answer or set about finding answers to such questions.

---
Torkel Franzen