[FOM] Re: Indispensability of the natural numbers

[Vladimir Sazonov]

Timothy Y. Chow wrote:
> On Tue, 18 May 2004,  David Isles wrote:
> 
>>Roughly speaking, as I understand it, the usual picture of the
>>"standard" natural numbers (closed under all primitive recursive
>>functions) is to the "true" natural numbers (given by rules R1, R2, and
>>R3) as the "nonstandard" model theoretic natural numbers are to the
>>"standard" natural numbers.
> 
> 
> Off-list, David Isles has clarified to me that "roughly speaking" was an
> important rider here, and that there isn't a known way to formalize "There
> exist N1 and N2 that satisfy R1, R2, R3, but N1 is closed under primitive
> recursive functions while N2 is not."

A theory of feasible numbers (based on an appropriate weaker version
of classical logic) which I already mentioned in posting to you gives 
the necessary example of (imaginary) N which is closed under +, but
does not contain 2^1000.  Did you read this place in my posting?

I also mentioned (actually, as I know, the first result of such kind
for the case of ordinary classical logic by Rohit Parikh, JSL 1971)
where a superexponential tower of 2s of the hight 2^1000 is not N.

Of course, these two versions of N cannot be represented as sets in
"the" universe for ZFC. But corresponding axiomatizations are
consistent.


> Tim

Vladimir