[FOM] First Order Logic/status

[Colin McLarty]

Harvey Friedman <friedman at math.ohio-state.edu> 
Tue, 24 Oct 2006 22:00:59 -0400 

Wrote:

> FOL (through its practically driven variants) is the only 
> appropriate vehicle for the foundations of mathematics. I 
> doubt if many people would disagree with this. The only 
> issue is how to go about carefully formulating just what 
> this means.

I want to say I entirely agree with this.  Harvey and I have notably 
disagreed over what kind of axioms in FOL ought to be taken as 
foundational.  But after thinking about it a whole lot for years, and 
seriously considering various options, I entirely agree that FOL is the 
vehicle.  For one example, precisely in my field of topos theory, the 
higher-order internal logic is itself a manifestation of more 
fundmental properties of each topos expressible in FOL.

I will not try to give my reasons -- because I also agree with Harvey 
that there remains a serious task of fully formulating the claim.  

best, Colin