[FOM] RE: FOM Continuum Hypothesis
Matt Insall
montez at fidnet.com
Fri May 16 14:15:32 EDT 2003
Mangin:
1) Do you believe that the continuum hypothesis is true, or false?
Insall:
False. The negation of CH allows us the freedom of assuming there are many
more cardinal numbers than CH itself allows. (CH ``collapses'' the
universe.) Also, I am a fan of extending to the infinite certain types of
properties of the finite. The following statement is true about finite
sets:
For n>0, there is a cardinal k>n such that the cardinality of the power set
of n is greater than k.
If this is adopted as an axiom for set theory and cardinal number theory,
then not only is CH false, but also, no version of GCH holds: for every
cardinal k, the successor of k is not 2^k.
Mangin:
2) Is there any general consensus amongst the mathematical/FOM community
regarding the truth or falsity of CH?
Insall:
I think several FOMers don't think the question of truth or falsity of CH
makes sense. I think some mathematicians believe CH is true. Thus, I doubt
that there can be said to be a consensus one way or another, or even on the
meaningfulness of the question. See for example, the the monthly article by
Solomon Feferman ``Does Mathematics Need New Axioms?'' Amer Math Monthly 106
(1999) no 2, 99-111. I don't want to put words in his mouth, but it seems
to me that Feferman believes the question of truth or falsity of CH is a
meaningless question, while the question of truth or falsity of other
questions we have not answered are not meaningless. For example, he
specifically mentions that he thinks the P vs NP problem is meaningful and
solvable. I think the question of whether CH is true or false is
meaningful, but that we may never solve it to the satisfaction of a majority
of mathematicians and FOMers. To understand my contention here, look at the
status of the axiom of choice (AC). There is a sense in which the problem
of the truth or falsity of AC has been ``solved''. Basically, mainstream
mathematics has adopted ZFC as the proving ground for mathematics, thereby
essentially declaring AC to be true. This is not a solution in the same
sense that the four-colour problem has been solved. It is a ``cultural''
solution. I think a similar kind of resolution can occur for CH, but only
if it is vigorously pursued. The axiom of choice has been so pursued, and
many mathematicians find themselves wanting to use the axiom of choice while
proving some theorem. The relative consistency results - such as
con(ZF)--->con(ZFC) - make it possible for such mathematicians to feel
reasonably comfortable using AC, which was once hotly contested, because it
leads also to certain anti-intuitive results. In a sense, mainstream
mathematics has weighed the pros and cons of AC and decided in favour of it.
CH, or even V=L, can be so studied, and are being studied this way to some
extent, but I think it is less common to find a mathematician who has used
either of these axioms than it is to find someone who has used AC in their
work. Sierpinski wrote a book on the consequences of CH, ``Hypothese du
Continu'' (1956), but I have not read all of it. However, some consequences
of CH are also some of the ``negative'' consequences of AC. For example,
Sierpinski gives a proof, due I think to Banach, that CH implies that there
is a nonmeasurable set, and this is done without AC (for otherwise CH is not
needed). If the ``positive'' consequences of CH seem to mainstream
mathematicians to outweigh, in some sense, the ``negative'' consequences of
CH, then perhaps CH will one day be adopted. But if only ``negative''
consequences of CH abound, perhaps mainstream mathematics will lean towards
an axiom to add to ZFC that implies the negation of CH.
Mangin:
3) What are the most important recent developments post-Cohen which have
contributed to this consensus (or lack thereof)? Have set-theorists proposed
any plausible axioms which might decide CH? Have any consequences of CH been
discovered which either strongly support or strongly undermine it?
Insall:
I do not think I can yet answer this question. Perhaps someone else on the
list can help with it. The example I gave above - that CH implies the
existence of a nonmeasurable set - does not really undermine CH, unless we
first decide to do without AC because of that particular consequence. I do
not see mainstream mathematics doing away with AC on that basis - at least
not anytime soon. I once gave my reason to this list for denial of CH, and
one response, if I recall correctly, was that the negation of CH leads to a
``dearth of functions''. There is a sense in which CH does lead to a
``simpler'' theory,so perhaps on the basis of ockham's razor, it will one
day be adopted widely. I think an approach that could be tried, but has not
received a lot of attention, is to try to connect CH or its negation to some
physical theory, and then perform physical experiments aimed at providing
support for the adoption of one or the other. For example, if we could show
that CH implies that some particular subatomic particle cannot exist, but we
perform experiments and these show that such a particle does exist, then we
would have physical ``proof'' that CH is false. It seems to me though that
the gap in understanding and communication between physics and fom are broad
enough that I will not see this done in my lifetime.
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