I am writing up a proof that Peano arithmetic (P), and even a small fragment of primitive-recursive arithmetic (PRA), are inconsistent. This is posted as a Work in Progress at http://www.math.princeton.edu/~nelson/books.html

A short outline of the book is at:

http://www.math.princeton.edu/~nelson/papers/outline.pdf

The outline begins with a formalist critique of finitism, making the case that there are tacit infinitary assumptions underlying finitism. Then the outline describes how inconsistency will be proved. It concludes with remarks on how to do modern mathematics within a consistent theory.

There's some discussion of this at The n-Category Cafe, including a brief (so far) discussion between Terry Tao and Nelson himself. Obviously, I expect a flaw will be found in his proof -- but I sure hope he's right! That would mean

*all of mathematics*will be in need of revision.

Update: Nelson says that Tao has indeed found a hole in the proof. Exponentiation remains total, for now.

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