FOM: status of CH
Randall Holmes
holmes at catseye.idbsu.edu
Tue Feb 3 17:07:59 EST 1998
This posting is from M. Randall Holmes
I'm not sure I understand why CH is supposed to be a problem for realism.
This is not to say that I don't understand why certain people think
that; but it seems to me that the argument(s) for its being a problem
are fallacious.
There are implicit assumptions behind the conviction of many on this
list that CH is a difficulty.
Assumption 1: Every mathematical question that has an answer must have
an answer that we can find.
Hilbert has been quoted on this list as asserting something like this.
(Incompleteness results might be taken to cast doubt on it, or might
not, but I'm not going to argue this point.) This is a very
implausible assumption from a realist standpoint; I see no reason to
believe that every question we can ask and understand about a reality
outside of us must have an answer that we can find. If mathematics is
regarded primarily as a human activity from the outset, then it might
be doubtful whether it made sense to say that a mathematical question
that people cannot answer (or even have not answered so far) has an
answer. But such a view of mathematics is not realist.
I find the axioms of ZFC to be a convincing description of part of
mathematical reality. They are not a complete description (even of
that part of mathematical reality). In particular, they are not a
complete description of the continuum.
Assumption 2: It is surprising that ZFC is not a complete description
of the continuum.
This is not very surprising at all. ZFC does not provide us with much
in the way of principles which would restrict how many subsets a
countably infinite collection (or any infinite collection) can have.
The major assumption it makes which might be regarded as making such a
restriction is the Power Set Axiom, which tells us that the subsets of
\omega (however many there are) can be collected into a set, and so
the number of subsets is small in relation to the size of the
universe. This is not much of a restriction!
V=L is a restrictive assumption which tells us quite a lot about how
many subsets the continuum or any infinite collection will have.
Apparently most set theorists do not believe V=L (I have met one who
professes to -- and who works on supercompact cardinals :-) )
Assumption 3: I have to be familiar with the extension of a set in order
to know what it is.
I don't think that my realist understanding of the continuum is
affected by my not knowing how many members it has, any more than my
understanding of what a swan is is limited by my not knowing how many
swans there are. I know a real number when I see one (or a set of
natural numbers when I see one), so I know what the class of real
numbers is. If I accept the Power Set Axiom, I have to believe that
the continuum is a set, and so small relative to the size of the
universe. But I don't have to commit myself (indeed, I have no
grounds for committing myself) to any particular size for the
continuum. The work of G\"odel and Cohen shows me that I'm right in
not committing myself (unless I am convinced by the proposal of V=L).
I suppose that there is another conclusion one could draw; perhaps the
results of Cohen point toward the conclusion that the Power Set Axiom
is false and the continuum is a proper class. If this is combined
with von Neumann's proposal that all proper classes are the same size
we get a very tidy foundation for mathematics with just two infinite
cardinals, both of which are part of the mathematical experience of
everyone. I don't believe this because I don't see any reason to
believe that the continuum is inexhaustible in the same sense in which
the (von Neumann) ordinals have to be. [the qualification is
necessary because there are foundational theories in which a "set of
all ordinals" is available] Notice that such a foundation would not have
to be weak in any sense; there still might be inner models in which there
were large cardinals. The conclusion would not be that the universe is
small; it would be that the continuum is enormous!
I don't have any idea how large the continuum is. I don't see any
particular reason why we should ever be certain how large the
continuum is. I do think that it is interesting to investigate the
various possibilities. None of this makes me doubt that there is a
fact of the matter. (The only situation in which I would doubt that there
was a fact of the matter would be if the universe were actually finite).
Assumption 4 (explicitly made by some on this list): a correct philosophy
of mathematics will assist one in producing mathematical results (and
its correctness should be judged on this basis).
This strikes me as rather doubtful. I don't think that being a
realist (though I am willing to argue that it is the correct position)
is going to assist me in proving theorems. I think that there are
some philosophies of mathematics which will positively handicap one in
proving theorems (any that cast doubt on the validity and vital
importance of rigorous proof as at least an ideal to be aimed at). I
think that there are incorrect philosophies of mathematics
(intuitionism, for example) which have directly inspired interesting
mathematical work.
And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes
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