FOM: What is the standard model for PA?

Torkel Franzen torkel at
Wed Mar 18 06:22:49 EST 1998

  Vaughan Pratt says:

  >It's indeterminate because you cannot prove from your definition the
  >following Property F:

  >F:	There are finitely many natural numbers between 0 and any given
  >	natural number n.

  As you will recall, I freely granted that my "definition" doesn't
even explicitly indicate that there is no end to the natural numbers.
In saying

       All the natural numbers are 0,s(0),s(s(0))
       and so on. What's unclear or indeterminate about this?

I did not mean that the phrase "0,s(0),s(s(0)) and so on" in any
formal sense defines the natural numbers. It only serves as a reminder
of what we're talking about, as part of the question what's unclear or
indeterminate about the natural numbers.

   Our understanding of the natural numbers isn't based on any formal
definitions - rather, we rely on that understanding in proposing formal

Torkel Franzen

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