FOM: Re: Does Mathematics Need New Axioms? (response to Matthew Frank)

[Matthew Frank]

I now respond to
 Simpson Thursday (May 25) responding to
  me Monday (May 22) responding to
   an already long debate;
this post is self-contained.

1. 
> [Simpson:] My impression is that the anti-foundation axiom has been
> of interest not so much for set-theoretic f.o.m. but in somewhat
> different contexts (Aczel's Frege structures, etc).

I agree.  I would further point out that people who work on logic and
games seem to like the AFA way of thinking.  In any case, if we need new
axioms for set theory, we might need those axioms so that set theory can
better serve some role other than that of foundation of math.

2.  Among variations of Steel's questions, Simpson has now asked:

> [Simpson:] Is this area of research likely to lead to new technology
> which can be expected to participate in improving the human standard
> of living?

I would answer no for most of f.o.m. and most of pure math.

3.
> [Simpson:] ZFC-style set-theoretic foundations has achieved a certain
> degree of acceptance among core mathematicians, which alternative
> foundational schemes have not achieved

The comparison between, say, ZFC and predicative mathematics seems a bit
unfair since ZFC (in roughly present form since the mid-20s) had a big
headstart over predicative mathematics (which I think was not in an
analogous state of development until at least 1960).  Most mathematicians
have not passed judgment on predicative mathematics; they simply haven't
heard of it, and I suspect that if they had heard of it many of them might
be sympathetic to it.  (By contrast, many mathematicians have heard of
constructive mathematics, and have evaluated it negatively.)

4.  [I think this topic is a new thread, and am posting it separately.]

5.  More about my post on Hilbert's second problem on the consistency of
arithmetic.  I think it will be easiest to quote what Wilfried Sieg wrote
me in response (and I hope he will not object):

> [Sieg:] You are quite right in stating that arithmetic is here NOT
> elementary number theory; Hilbert refers explicitly to his paper "Ueber
> den Zahlbegriff" in which an axiomatization is given for the reals.
> (More generally, even in the early twenties, Hilbert and Bernays use
> "arithmetic" in a very broad sense, sometimes even including set
> theory.)
> However, I don't think your "precise version" of Hilbert's question is
> anymore a version of Hilbert's question.  One way of making Hilbert's
> question precise is to consider full second order number theory.  That
> is perfectly in line with Hilbert's earlier (unpublished) considerations
> and the later work, beginning in the lecture notes from 1917/18.

More soon....

--Matt