[FOM] Tolerance Principle

Thomas Forster T.Forster at dpmms.cam.ac.uk
Thu Feb 9 11:59:12 EST 2006

That doesn't really address my concern.  Bill Taylor was asking me about
the particular case of NF and ZF.  This reminded me of an old puzzle about
the intersection (one might call it `NZF') of these two theories.  The
intersection cannot be proved consistent in both of them, because of the
second incompleteness theorem.  Or so one would expect. To apply this
insight straightforwardly one needs the two theories to interpret
arithmetic in the same way, which they don't.  (My guess is that ZF proves 
Con(NF) and a fortiori Con(NZF) too but that is by-the-by.)  Any helpful 
apercus about the significance of varying interpretations of arithemtic in 
this context would be gratefully received.

On Wed, 8 Feb 2006, Harvey Friedman wrote:

> On 2/8/06 4:18 AM, "Thomas Forster" <T.Forster at dpmms.cam.ac.uk> wrote:
> > 
> > 
> > 
> > I think the two systems have to interpret arithmetic in the same
> > way......  (which NF and ZFC do not)
> > 
> >    tf
> > 
> > 
> The question of whether or not two appropriate systems interpret arithmetic
> in the same way has NOTHING to do with the question of whether one of the
> systems is interpretable in the other.
> E.g., a lot of (weak) set theories are interpretable in a lot of arithmetics
> (systems of arithmetic), and vice versa.
> Harvey Friedman 
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