[FOM] Consistency of Peano Arithmetic

[Daniel Mehkeri]

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.

Frode Bjørdal has answered by referring us to Edward Nelson's work. But 
Nelson is of course an ultrafinitist, so this just underlines what I am 
saying. The rhetorical space between the elementary recursive and the 
Gamma_0-recursive is awfully quiet!


Daniel Mehkeri