FOM: What is the standard model for PA?

Torkel Franzen torkel at
Thu Mar 19 05:07:35 EST 1998

  First a response to Vaughan Pratt:

   >Are you contradicting this, or claiming that our understanding of
   >finiteness is determinate?

  Finiteness is as determinate as the natural numbers, sure.

  Vladimir Sazonov says:

  >From the point of view of the above discussed illusion, when 
  >we concentrate *only* on it, we will hardly see anything 
  >unfixed.  It comes in mind the "full" induction axiom on 
  >"arbitrary" properties of natural numbers which would fix the 
  >standard model completely (up to isomorphism) as in the 
  >framework of set theory. You know that the term "arbitrary" is 
  >too problematic here. It bothers me too much to consider this 
  >model as "really" fixed.

  I quite agree that the notion of "arbitrary property of natural
numbers" is indeterminate, and I don't at all think that the full
induction axiom for arbitrary properties of natural numbers can be
invoked to establish the determinateness of the natural numbers.
This does not imply that there is anything indeterminate or unclear
about the natural numbers, or the quantifier "for every natural

  >Thus, we have many different infinite 
  >natural number series. I do not see how we will get "all" 
  >natural numbers in this process even potentially.

  The line of thought that you summarize as leading to different
infinite natural number series is one that may well be worth
pursuing  (although I admit that I don't myself put much faith in
Yesenin-Volpin's consistency proof for ZFC). "All numbers" does
indeed become an indeterminate notion on this line of thought.

   >In particular we implicitly 
   >learned at school some (idea of) induction or iteration rule, i.e., 
   >essentially Peano Arithmetic, which allows us to *deduce* easily 
   >that exponential (= iterated multiplication) is total function 
   >and logarithm is bounded.

  Yes, but the *justification* for these arguments is not found in
axioms, but in our understanding of the concepts involved. Also,
I believe that for very many people it's a lot easier to recognize
that addition is commutative than it is to recognize an inductive
proof of the commutativity of addition as a proof.

  >Who do not know what is *physically* written 
  >string in a finite alphabet?

  The notion of a physically written string is full of uncertainties
and indefiniteness. What is required for an arrangement of chalk dust,
pencil lead particles, and so on, to constitute a written string in
a particular finite alphabet?

Torkel Franzen

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