FOM: Upcoming Panel Discussion
Matt Insall
montez at rollanet.org
Fri Feb 18 01:24:10 EST 2000
First, I would like to thank Professor Simpson for helpful comments about my
original post.
I am quite happy to see the upcoming panel discussion about whether new
axioms are needed in Mathematics. Let me first comment that I do know some
Mathematicians who find no use for Foundational issues, especially the
axiom
of choice or the continuum hypothesis. However, there are those who would
like to see a more ``solid'' resolution to the issues than just ``It's
consistent, but independent.'' It is my hope that this century, aye,
preferably my lifetime, sees such a better resolution to these problems.
Please be patient as I bring my views and small bit of understanding to
this
discussion. Firstly, I was very glad to see Professor Friedman's posting
which outlined his understanding of the various views. Secondly, I was
pleased to see the lively discussion that ensued. I ``join the fray'', as
it were, in hopes that I can contribute something of interest. I'd like to
consider each of the panelists' views individually, if I may.
Professor Maddy:
Professor Maddy is correct, IMHO, in considering the continuum hypothesis
to
be of great interest in and out of Mathematics. I have seen significant
amounts of discussion about it, in spite of the number of scientists and
mathematicians I meet who seem not to know the statement of the problem.
(This is, presumably due to a lack of interest, or a lack of time to devote
to anything outside their own subject area.) The basic problem that we
cannot decide the continuum hypothesis from only the axioms of ZF (and some
other) set theories has seemed to be a stumbling block for many. It is
interesting to me that Professor Maddy makes a claim I had not heard made
by
FOMers before (at least not those who have seen Gödel's second
incompleteness theorem), namely that ZFC is consistent. This seems
somewhat
bold, unless what was meant was ``relatively consistent''. In any case, I
agree with Professor Maddy that new axioms are needed to settle the
continuum hypothesis. Also, as is ascribed to her by Professor Friedman, I
see no need to base this need for more axioms on the consistency of ZFC.
For if even ZF were inconsistent, we would presumably restrict our
attention
to some provably consistent fragment of ZF, and begin trying to complete
that new axiom system. Such an attempt to complete the ``new set theory''
would include a need to answer (at the least) a ``continuum
hypothesis-like'' problem. It is stated that many mathematicians are not
interested in the continuum hypothesis, and that Professor Maddy is aware
of
this fact. However, I contend that there are mathematicians who are
interested in certain results which are entailed by the continuum
hypothesis. For an example which is closely related to modern results on
measurable cardinals (see Jech, pgs. 295-397), I would expect that the
following result of Banach and Kuratowski [Fund. Math. 14(1929), 127-131
(in French)] should hold some interest for analysts and probability
theorists:
``There is no real-valued function m defined on the power set of the
interval [0,1] such that
(1) If X is a singleton, then m(X) = 0
(2) m is countably additive (i.e. if X_1,X_2,... is a countable
family of pairwise disjoint sets in [0,1], and if X is their
union, then m(X) = m(X_1) + m(X_2) + ...
(3) m is not identically zero.''
This result is somewhat improved in the book by Sierpinski by the title
``Hypothese du Continu'', where he shows that the result still holds if the
interval [0,1] is replaced here with any set of reals. His argument is
based upon the continuum hypothesis. In fact, in the original paper by
Banach and Kuratowski, the last page contains a remark that this result
(that there is no continuous, countably additive measure on the unit
interval [or, according to Sierpinski's presentation, on any set of
reals]),
together with an apparently mild lemma they used in the proof, yields the
continuum hypothesis as a consequence. I'll state the ``mild lemma'' in
French:
``Etant donees deux suites d'entiers positifs S = {k_i} et T = {n_i},
convenons d'ecrire T < S, lorsque n_i \le k_i quel que soit i. Il existe
une famille F, de la puissance du continu, ayant comme elements des suites
d'entiers positifs et telle que, pour chaque suite S (qu'elle appartienne a
F ou non), l'ensemble des suites T de F telles que T < S est au plus
denombrable.''
(I have not tried to show that this lemma alone implies the continuum
hypothesis, but I doubt it.) Thus a very practical (from an analyst's
point
of view) mathematical consideration, namely, ``Is there a continuous
measure
on some set of reals.'' hinges on the fate of the continuum hypothesis.
(Of
course, by extension, the question of the existence of a continuous
probability measure is also jeopardized by the continuum hypothesis as
well.
To some who do not think analysis and abstract measure theory are very
practical, the problem of assigning probabilities should seem even more
practical.) Moreover, even mathematicians who do not pay a large amount of
attention to the continuum hypothesis may not be entirely
``disinterested''.
Consider ``Fermat's Last Theorem''. Many mathematicians worked on this
problem or related problems, but also, many just worked on other things.
The fact that its purported resolution by Wiles caused such a stir
indicates
that many mathematicians who may not have appeared very interested in the
problem were indeed interested, but chose to pursue other interests more
ardently. Along these lines, I wonder what the result of a more generally
distributed survey like the one posted to this list by Professor Jim Brown
would be. My own responses are as follows:
A. How much do you know about V=L, MC, etc.? My answer is: 2. Read some
literature, perhaps teach a course in which these issues arise,
B. What are your philosophical views? My answer is: Platonist.
C. Which axiom do you accept? My answer is: 3. Other or no view on the
matter.
I would comment as follows: I have reason to accept the following
restricted version of Professor Maddy's ``MAXIMIZE'' principle: ``There
are
as many cardinals between \aleph_0 and c as is logically possible.'' (It
seems to me that there are then c such cardinals, although I've not written
a
formal argument along these lines, and I presume that the ``large
cardinal''
axioms already include this one somewhere.) Thus I come down strongly
against the continuum hypothesis (and by extension, against V=L). However,
I do not abandon the possibility that some work I am doing now with some
colleagues here at UMR, or some arguments presented by other mathematicians
and scientists, would convince me otherwise.
Name: Matt Insall
Position: Associate Professor of Mathematics
Institution: University of Missouri - Rolla
Research interest: Foundations of Mathematics
More information: http://www.umr.edu/~insall
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