[FOM] Consistency of Peano Arithmetic

Carl Mummert mummertc at marshall.edu
Fri May 13 09:06:36 EDT 2011

In a message to FOM dated May 11, 2011 at 9:24 PM, Harvey Friedman wrote:
>  I asked very specifically whether [t]he consistency of Peano Arithmetic is a legitimate mathematical problem in present day mathematical culture.


> I am shocked, because there is a seemingly perfectly understandable and normal mathematical proof of the consistency of PA which is entirely within the current fabric of mathematical practice.

I'm glad these remarks were posted to the FOM list - I have seen this
question elsewhere recently, and I agree it is an important question
in contemporary foundational work.  One of the motivating questions of
FOM is "what is a proof", and the consistency proofs of Peano
arithmetic and Heyting arithmetic are key test cases.

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?

- Carl Mummert
Marshall University

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