FOM: {n: n notin f(n)}

[Harvey Friedman]

As Buckner and Heck (and others) continue their back and forth, I 
would like to make a few comments.

1. As far as foundations of mathematics is concerned, the set 
theoretic interpretation of mathematical statements and mathematical 
proof is the currently accepted standard with no peer. There are 
other approaches, such as  restrictions on the set theoretic 
approach, or categorical foundations. The latter does not, as least 
yet, seem to have status independent of set theory, but even if one 
insists that it does, the current formulations are immediately 
mutually interpretable with set theory.

2. This set theoretical interpretation of mathematics is extremely 
coherent, and totally natural, especially if one restricts to the 
level of set theory that is comfortably sufficient for the great 
preponderance of mathematics. I am thinking of, say, Zermelo set 
theory with the axiom of choice (ZC). Of course, it is of great 
interest to see in what way one can or cannot get away with much 
weaker set theoretic commitments than ZC.

3. In fact, set theory and its fragments appear to be the only fully 
rigorous totally natural completely coherent wide ranging powerful 
systematization that we have throughout the whole intellectual 
landscape. Its rivals, e.g., Newtonian mechanics, Bayesian 
statistics, Euclidean geometry, religious doctrine, etc., fall apart, 
in comparison, on the fully rigorous side and/or on the wide 
ranging/powerful side. There is nothing else like it, and I regard it 
as a major challenge to develop something like it in a different 
context. I think this can be done in an effective and impressive way 
in many contexts, but it so far has proven too difficult for people 
to accomplish. We are not even at the point where people realize just 
how vitally important it is to accomplish this, and what is to be 
gained by it.

4. The kind of attitudes and objections being raised in the FOM 
discussion have not, at least yet, been shown to be any relevance or 
usefulness for the foundations of mathematics - or even foundations 
of science generally. In comparison with the great systematizations 
for f.o.m., these attitudes and objections are merely incoherent, 
useless, vague, and arbitrary - at least in their present form.

5. It is the burden of those who hold such attitudes and objections 
to develop something substantial out of their attitudes and 
objections. If these attitudes and objections are not intended to be 
relevant to f.o.m., or the foundations of science, then that should 
be stated at the outset.

6. Also relevant is something I have said earlier on the FOM:

The progress of mathematics and science has necessarily lead to the 
consideration of concepts that do not coincide with any that are 
commonly used in ordinary informal discourse. For instance, 
physicists and chemists do not convey their discoveries in terms of 
"fire" and "hot" and "wet" and "dry". It was only through great 
progress that "water" was, to a substantial extent, preserved.

It is not clear whether mathematics (ultimately formulated in set 
theoretic terms) has an interpretation in ordinary informal 
discourse. I have conjectured that even set theory with large 
cardinals has a convincing interpretation in
ordinary informal discourse, sufficient to prove its consistency 
(freedom from contradiction) within ordinary informal commonsense 
reasoning. However, it appears that we are some distance from 
establishing that conjecture.

The work I presented in my postings #155-157 is intended as a step in 
this direction.