FOM: What is the standard model for PA?

Torkel Franzen torkel at
Mon Mar 16 03:40:42 EST 1998

   Validimir Sazonov says:

   >On the other hand it is not me who asserts determinacy or
   >clarity about the notion "natural number". It seems that my
   >position is more safe: If somebody asserts this, let him explain
   >at least what does it mean.

  "Determinate", like "real", is what J.L.Austin called a "trouser
word": it's the the opposite (or a range of possible opposites) that
wears the trousers. I can only say that I don't see anything
indeterminate or unclear about the notion "natural number", and ask
you to point out to me where you see an unclarity or indeterminacy.

  In your comments, you remark that "and so on" doesn't even say
that there is no last natural number. Sure, but you're not saying
that there is any unclarity about whether or not there is a
last natural number. You know there isn't, just as I know it. So
no unclarity on that score. You also point out that "and so on"
might cover only x for which log log x < 10. Sure, but you're quite
as aware as anybody else that on the ordinary understanding of
"natural number", log log x < 10 doesn't hold for every x. So there's
no unclarity or indeterminacy there either.

   >I do not know what else justification you need.  Experiments
   >like in physics (and the lack of counterexamples) are also some
   >kind of justification.  Of course, it is also necessary to
   >demonstrate mathematical and applied usefulness of such an
   >"arithmetical low" or illustrate how it works.

  I don't think it's necessary to demonstrate the mathematical and
applied usefulness of a proposed arithmetical law in this context, but
only to give some explanation of why one should believe it to be
true. The notion of "feasible number" may be a very useful and natural
one to invoke in many contexts and applications. But this affords no
grounds whatever for the claim that the ordinary notion of natural
number is in any way unclear or indeterminate, or that experience
should prompt us to adopt "for all x, log log x <10" as an
"arithmetical law".

Torkel Franzen

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