FOM: What is the standard model for PA?

Torkel Franzen torkel at
Tue Mar 17 03:14:05 EST 1998

 Validimir Sazonov says:

   >You don't see anything indeterminate or unclear about the notion 
   >"natural number" or you see that and how it is determinate?

  I don't see anything indeterminate or unclear about the notion
"natural number". "Seeing that it is determinate", as far as I can
determine, amounts only to not seeing anything indeterminate or
unclear about it. This doesn't exclude the possibility that you might
convince me that there is something indeterminate or unclear about

   >No, I do not know. More precisely, I know that I may consider both
   >alternative, each being reasonable in its own way. Let us fix that
   >there is NO last natural number because this is usually *postulated*
   >for "the standard model". This still does not fix this standard
   >model, just how *long* is it.

  "The standard model" is jargon. You knew everything that I know about
the natural numbers (from the present philosophical point of view) long
before you had heard of "the standard model". One of the things you
learned was that there isn't any largest natural number - given any number
k, there is a larger one, k+1. It's not clear to me what you consider

  >Yes, I do know that it [for all x, log log x < 10] does not hold
  >according to such and such axioms.

  Axioms are secondary. People who have never heard of the Peano
axioms and wouldn't be interested in hearing about them learn and
apparently accept that log is an unbounded function, on the basis of
their informal understanding of "natural number". Is there some
confusion or unclarity in this?

  >If you say "all", I do not understand what does this mean. It is unclear 
  >what is "all" EVEN in the more reliable case of feasible numbers which 
  >have a physical, perceiving meaning in our real world.

  Then we're back to the beginning. All the natural numbers are 0,s(0),s(s(0))
and so on. What's unclear or indeterminate about this?

  As for the "feasible numbers", I don't know what you mean by "the
more reliable case". The notion of "feasible number" is very unclear
and indeterminate - isn't that so?

   >I have some llusion of understanding.

  Well, I would say that you rather have an illusion of not
understanding.  Or, in less confrontational terms, I would say that
you have certain ideas about what is required to understand something,
or what is required for something to make determinate sense. I am
certainly not in a position to say that these ideas are wrong, that
you are mistaken, that your pursuits are misguided. On the contrary, I
think it's a good thing that people pursue such ideas and
inclinations, including stuff like Yesenin-Volpin's consistency proof
for ZFC (or NF). Where I think you are plain wrong is only in
presenting your ideas and inclinations as though they revealed some
defect in standard practice.

Torkel Franzen

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