[FOM] Consistency of Peano Arithmetic

[Frode Bjørdal]

2011/5/13 Carl Mummert <mummertc at marshall.edu>:
> 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.
>
> and
>
>> 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?

Here Edward Nelson's work
(http://www.math.princeton.edu/~nelson/books/pa.pdf) merits awareness.

-- 
Frode Bjørdal
Professor i filosofi
IFIKK, Universitetet i Oslo
www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html