FOM: What is the standard model for PA?

Vaughan Pratt pratt at cs.Stanford.EDU
Tue Mar 17 12:12:04 EST 1998

From: Torkel Franzen <torkel at>
>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?

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.

Granted that you cannot expect the man in the street to know about
nonstandard models.  But you *can* expect him to know property F.  That he
doesn't know about nonstandard models does not prevent the experts from
using a nonstandard model such as the ordinals up to omega+omega to prove
that the man in the street cannot infer property F from your definition.

So when the man in the street asks you how property F follows from your
definition, it is your responsibility to answer.  It would not be fair
to make him prove that you can't answer.  But it would be fair to let
him ask the experts, and we will tell him that your definition does not
entail that property and therefore is incomplete.

And then he might be motivated to ask us how we proved it.  Logic is
a fit subject for high schools when explained in the right order.
I was overjoyed when my high school swim coach lent me his logic text
from college.  That it was only about Aristotelian syllogisms did not
bother me one bit, but if it had started out by showing that all Boolean
rings were Boolean algebras I might have been less interested.

Vaughan Pratt
				Poke not fire with a sword.  --Pythagoras

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