FOM: Significance of work on Goedel's program

[Solomon Feferman]

Steve Simpson's intemperate reaction to the recent spate of messages on
the status of CH as a mathematical problem is understandable.  The
discussion seemed to have reached a point of non-return, or at least to be
circling around the same issues over and over, and conclusions on which
there could be common agreeement did not seem to be emerging.  So better,
then, to turn to specific problems (specific within an agreed-upon
context) which appear to be relevant to f.o.m., where at least one can
hope to obtain definitive progress.  At least that was his suggestion.

But the source of the discussion was simply one opinion which I advanced
in my paper "Does mathematics need new axioms?" ([NA?] below), beginning
with a sympathetic summary by Torkel Franzen on 11/14 and support, to an
extent, by Simpson on that same date.  The point of [NA?] was to assess
the logical work on Goedel's program for new axioms to settle undecided
arithmetical statements and undecided set-theoretical statements,
especially CH.  Much work in logic speaks for itself, e.g. the solution of
Hilbert's 10th problem, or model-theoretic algebra, or even NSA.  Much of
set-theoretical work also speaks for itself.  The work can be reported to
possibly interested audiences without trying to plumb its significance.
But that does not seem to me at all to be the character of the work on
Goedel's program, and there the hard part is trying to get at its
significance.  Indeed, that seems to me to be required with all sorts
of work that appears to be in the f.o.m., whether it be in set theory or
proof theory or constructivity or whatever.  Perhaps, in doing so,
individuals will arrive at irreconcilable viewpoints, but hopefully the
process of sharpening these positions is progress of a sort.  At the very
least, as Robert Tragesser has said, it can be "stimulating" and
"bracing".  

The paper [NA?] is directed at a general mathematical audience, and so for
that reason also, it could not go into more detail about the logical work
at issue.  I think it is important for mathematicians at large to give
serious thought to foundational questions but at the same time to realize
that not anything goes, that there has to be some cognizance of the
substantial amount of work that attempts to address these questions, and
to give some indication of what that work is.  At the same time, I think
it is the responsibility of logicians who are concerned to bring the
attention of mathematicians at large, to work on results of foundational
relevance, to exercise some care in explaining the significance of that
work.  

The assessments of work on Goedel's program that I arrived at in [NA?]
were well summarized by Franzen in his posting of 11/14, but let me repeat
them briefly:
1. Are new axioms needed to settle undecided arithmetical problems?  It
depends whether you are talking about (a) previously formulated problems
such as Goldbach's conjecture, twin primes conjecture, Riemmann
Hypothesis, etc., or (b) new, more or less perspicuous arithmetical or
finite combinatorial problems that have been connected with large cardinal
hypotheses.  My assessment in the case of (a) is that there is no evidence
on the basis of current logical work of the need for new axioms to settle
such problems, and in the case of (b) that what we need to know about
those large cardinal hypotheses is not their truth but only their
consistency (or 1-consistency).  It is misleading in the case of (b) to
claim that if a certain combinatorial statement is equivalent to the
(1-)consistency of a large cardinal hypothesis, that this shows we "need"
that hypothesis as a new "axiom".  This is a clear case of
question-begging.  
2. Are new axioms needed to settle CH?  In this respect I reported the
dazzling work by set-theorists in the last 30-35 years on proposed new
axioms and what might be reasons (intrinsic or extrinsic) for accepting
them, and stated (as Tony Martin had done in his assessment for the
Browder Hilbert-problem conference) that despite all the progress that has
been made, CH has been shown to be independent of all remotely plausible
axioms of infinity that have been considered so far (assuming their
consistency).  I then went on to raise my personal doubts, based on
philosophical considerations (that, yes, go back to gut feelings), as to
the very presumptions of Goedel's program vis a vis CH, namely the
presumption that CH is a definite mathematical problem.
3. One of the reasons for believing that CH is a definite mathematical
problem lies in the common belief that the continuum is a well-determined
mathematical structure.  Arguments for that come from both a platonistic
direction (Goedel, supremely) and a naturalistic direction (Quine,
Putnam, among others).  Since I reject platonistic metaphysics, what can
be said of the latter?  This resonates with a common view that the
continuum is "in nature".  The Quinean argument is that mathematics is
indispensable to science, our most reliable source of sure knowledge, and
thus science justifies mathematics.  So to assess this argument, we need
to see just which part of mathematics is justified by scientific
applications.  And here the logical work has come to the conclusion that,
at least as far as foundational assumptions are concerned, a little bit
goes a very long way: formal systems conservative over PA suffice to
formalize all known mathematics needed for currently generally accepted
such applications.  But that does not deny the possibility of the need for
stronger foundational principles in future applications or the great
instrumental value of various kinds of higher set theory even for current
applications.  But philosophically, the naturalist position on the
continuum is not supported by the logical work.
4. My conclusion from 2 and 3 was that CH "is an inherently vague problem
that *no* new axiom will settle in a convincingly definite way".

It is 4, and in particular my use of the words "inherently vague" as
applied to CH that aroused the most controversy, and to which Steve
reacted.  I will not try to summarize the arguments; as Robert Tragesser
said, there is a thread that just has to be followed.  But I will point to
the postings of Torkel Franzen as at least elaborating what I have in mind
here, and recommend their re-reading.  I don't want to continue the
discussion unnecessarily, if indeed people feel we need a rest from it,
but let me just add a few remarks by way of clarification on my own.
Perhaps the use of the term "vague", with its considerable established
position as a topic of some importance in philosophy and natural language, was
mistaken.  And one might then ask: how much more than vague can something
be if it is "inherently" so?  Franzen suggested, by way of clarificational
substitution, "essentially indefinite".  My view is that we cannot say
what P(N) *is*, as a definite mathematical object, and that the quantifier
"all subsets of N..." is vague in a way analogous to "all bald men..." or
"all stars in the universe...".  The "inherently" attaches to my
expectation that nothing more we can try to say to determine it as a
unique mathematical object will serve to do so.  So, "essentially
indefinite" if you prefer, but I still think "inherently vague" is a
reasonable appelation for the point of view, and is a bit punchier.

Two criticisms are lurking in the background, one spoken and one unspoken.
The former is that the kind of negative assessements I have made are a wet
blanket meant to discourage work in certain directions.  On the contrary,
they are meant only to encourage rethinking of what the work actually
accomplishes.  The second, unspoken criticism, perhaps, is that one should
not tell mathematicians that they can ignore logic.  On the contrary, as I
said above, I want them to know that there is serious work in logic that
they have to take into consideration if they are going to form views on
foundational matters.  I would encourage those who oppose my assessments
to bring their own perspective, with the supporting evidence, to the wider
mathematical public as well.

Sol Feferman

PS. I "understand" what it means to be a set, what it means to be a
one-one correspondence between sets, what sort of set P(N) or, for that
matter any P(A) is supposed to be, and I can reason quite confidently,
without benefit of axioms, on the basis of those understandings.  So CH
certainly presents itself to me in my understanding as a meaningful
mathematical proposition, though not, as I have argued, a definite one.
If it had been settled in ZFC, it would not have become more definite
thereby. The same applies to all the results about the continuum or
various kinds of subsets of the continuum (e.g. Borel sets) which *have*
been settled in ZFC and in which P(N) is essentially involved.