[FOM] consistency of PA?

[Roger Bishop Jones]

I'm no expert on this topic, but I can come
up with a short answer to one of Randall's questions.

Randall Holmes asks:

> 1.  is it even possible for ~Con(PA) to hold?

Yes, but not in any language which gets the semantics right.
i.e. only in "nonstandard" models of arithmetic or weak set 
theories.
(i.e. not *really*, unless of course PA were inconsistent!)

> Can one form an intuitive picture of what the world would be
> like if PA were not consistent?  It seems uncontroversial that
> N is there: if we have an object to be 0 and a universally
> applicable "successor relation" (both are available in
> predicative set theories) it would seem that we should be able
> consistently to contemplate the intersection of all inductive
> sets (or even the intersection of all inductive classes, if
> our set theory is known not to have a set which is the
> intersection of all inductive sets.)  If the intersection of
> all inductive classes is a class and we can define (proper!)
> classes in a suitable way, then there is a copy of N.

In the models of set theory in which arithmetic is non
standard the real N is not a set.
All the natural numbers are there, but there is no set
which contains ONLY the natural numbers.
The non standard models are all ill-founded, and all
the sets which contain all the natural numbers also
contain copies of an ill-founded collection which looks
like Z, and include "numbers" which look like
the codings of derivations of contradictions in those
consistent formal systems whose consistency is not
provable in the relevant set theory.

Roger Jones

- rbj01 at rbjones.com
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