[FOM] Consistency of Peano Arithmetic

[Richard Heck]

On 05/16/2011 11:54 PM, Daniel Mehkeri wrote:
>
> Carl Mummert writes:
>
> > Of course one can argue that the consistency question is ill posed,
> > for example by adopting some form of finitism.  I am more interested
> > in contemporary arguments (preferably from those who make them, if
> > they have put them in writing) that accept the existence of the
> > usual set of natural numbers, and accept the usual methods of
> > contemporary mathematics, but doubt the consistency of PA or HA.
> > Would someone be willing to summarize these for the FOM list,
> > or provide references?
>
> Are there such people?
>
> For that matter I don't recall even a finitist position being 
> advocated here on the FOM. I know Bill Tait is on this list, and he 
> has a well-motivated analysis of finitism, but as I recall it is an 
> analysis from the outside.
>
> Doubt about HA or PA seems to come from ultrafinitist, ultraformalist, 
> or generally skeptical points of view. The nearest to that seems to be 
> the constructivist and/or predicative points of view, from whom HA or 
> PA are not in question at all. That's quite a gap.
>
I hope to find time to write about this issue in some detail later, but 
for now I'd like to register just one point: The phrase "doubt about PA" 
would seem to suggest doubt about the *truth* of all the axioms of PA. 
That is, prima facie, very different from doubt about PA's consistency. 
To take a slightly different example, one could perfectly well be 
skeptical about the *truth* of all the axioms of ZF without necessarily 
having doubts about their consistency. No less a logician and 
philosopher than George Boolos once held precisely that view.

And of course these obviously come apart in other places. I have no 
doubt myself about the consistency of PA + ~Con(PA).

In that sense, then, references to (ultra)finitism aren't necessarily 
helpful here.

Richard Heck