[FOM] Remedial mathematics?

[Arnold Neumaier]

Walt Read wrote:
> To extend the analogy a little, the result of the work on non-standard
> (non-Euclidean) models of geometry was a recognition of the other
> models as equally valid. Eventually it was considered reasonable that
> even ``the" universe might better be modeled by one of the
> non-standard models. Do we see N the same way? 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"? 

It is impossible to use any of the nonstandard natural numbers for 
counting - only the standard subset of natural numbers is being used for 
that.

This distinguishes ''the'' natural numbers from nonstandard models of 
the Peano axioms.