[simpson@math.psu.edu: Re: FOM: the exaggerated claims of topos theory qua f.o.m.]

[Colin Mclarty]

Reply to message from simpson at math.psu.edu of Sun, 18 Jan
>
      
>In set theory, the picture is, sets.  In topos theory, the picture is
>-- what?  There are too many wildly differing models for this to be a
>viable educational and/or foundational option.
      
      	I think this is the central difference. Set theoretic
foundations aim at one rather narrowly described picture (though of
course there are variants), categorical foundations have many quite
different variants. I think the range of variability, the organic 
unity of that range, and the different relations of different variants 
to mathematical practice, are advantages. Simpson thinks it makes
categories foundationally inviable. We may well be at bedrock here.
      
      
> > Compare Harvey's surprizing work on math in finite initial segments
> > of the cumulative heirarchy. Every topos (even without natural
> > numbers) includes equivalents to every finite initial segment of
> > the cumulative heirarchy. In any topos with natural numbers these
> > equivalents relate to the real numbers just as they do in sets. So
> > you can see far more than enough math is available for building
> > bridges.
>
>So now it emerges that the basis of McLarty's claim is finite initial
>segments of the cumulative hierarchy.  The full set of natural numbers
>isn't even needed!
      
      	This is the Star Trek effect. I make one point about the 
strength of some topos axioms by comparing them to something Simpson
knows. So he concludes this comparison is "the basis" of all my
claims.
      
      	I expect Simpson is right that this thread has run its course.
      
Colin