Edward Nelson at Princeton recently told his colleagues that he was

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 [here]

Peano arithmetic is arithmetic with natural numbers.

A short outline of the book is (here).

So the natural numbers are “inconsistent”?

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.

However, he has withdrawn his criticism based on this response by Terrence Tau, pointing out an error in his work.

In other words, there is such a thing as being correct or incorrect, not merely in power as opposed to out of power. Like some areas in science – that are not really “disciplines” – we can all think of.

(Physics is still a discipline in science because even Einstein might be wrong. Math is still a discipline because people can even admit the idea didn’t work and then just go back to their desks.)

The link described as Tau’s response leads to something else.

Nelson’s criticism may have been incorrect, but does that necessarily imply that his conclusion is incorrect?

It’s not the natural numbers that are ‘inconsistent’, it’s that the system of equations using Peano arithmatic would be ‘inconsistent’ in the mathematical sense (the system of equations would not have a solution)

an attempt to provide a definition for primitive recursive arithmetic?

Paranormal studies, homeopathic medicine, Lysinkoism…