[FOM] PA inconsistencies

[Alasdair Urquhart]

In reply to Harvey's question,

> FOM Readership: what are the main references for "mathematics without
> exponentiation" and how far has it gotten?

there is a large literature on bounded arithmetic and related questions.

For an excellent semi-popular account of such matters, I strongly
recommend Pavel Pudlák's expository book "Logical Foundations of
Mathematics and Computational Complexity."  Chapter 6 contains
a lot of material relevant to the questions under discussion.
If you are interested in bounded arithmetic, you can follow
up the references given there.