[FOM] PA inconsistencies

Harvey Friedman hmflogic at gmail.com
Sun Aug 19 12:38:55 EDT 2018

I meant an depth development of actual mathematics, (analysis,
algebra, number theory, differential equations, topology, geometry,
group theory, etc,.), not computational complexity developments or
math logic developments.

If you still regard your answer as responsive, it would be nice to
give some more information ahead of subscribers tracking down

I am aware of some developments, running into major open problems,
seeing what can be proved in bounded arithmetic, but that has a narrow
focus, and I was wondering about broader investigations.

In the realm of constructive analysis, there are people, most notably
Errett Bishop, who rolled up their sleeves to actually did a lot of
actual mathematical developments within its philosophy. And it was
highly systematic, and covered a huge amount of topics. I was looking
for this sort of POSITIVE thing, and if it really has not gone very
far, perhaps people should take it farther.

Harvey Friedman

On Sat, Aug 18, 2018 at 12:09 PM, Alasdair Urquhart
<urquhart at cs.toronto.edu> wrote:
> 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.
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