[FOM] Remedial mathematics?

[Dana Scott]

On Thu, 26 May 2011 Walt Read <walt.read at gmail.com> asked:

> Is there a unique thing
> denoted by ``the natural numbers", accessible to us through intuition
> or however, with the non-standard models of PA being essentially
> artifacts of the formalization process? Or do the non-standard models
> have equal claim as models of ``the natural numbers"? Might we at some
> time in the future see one of the non-standard models as better suited
> to our understanding as we learn more about natural numbers? The
> latter view would make N something of a work-in-progress and would fit
> well with the ``analogous (to some degree) with physical theory" view
> of FOM. But it would seem to be a very different view than most people
> have traditionally held. What is the position of classic FOM on this?

Let us recall:

TENNENBAUM'S THEOREM: There is no recursive non-standard model of {PA}.

FEFERMAN'S THEOREM: There is no arithmetically definable non-standard model 
			of all true sentences of arithmetic.   

In other words, there are no models just lying around in the closet
waiting to be used.  The situation is so much different in Geometry.