[FOM] CH and mathematics

Rupert McCallum rupertmccallum at yahoo.com
Thu Jan 17 02:48:11 EST 2008

--- Alex Blum <blumal at mail.biu.ac.il> wrote:

> In his recent entry in Philosophy of Mathematics:5 Questions, Solomon
> Feferman asks: "Is the Continuum Hypothesis a definite mathematical 
> problem?"
> Can someone elaborate on what the choices are or could have been?

Do you mean you want us to run through the different philosophical
positions and say whether they would answer yes or no to this question?

Or do you want to know what the possibilities are regarding the status
of the continuum hypothesis?

Assuming ZFC is consistent, there are three mutually exclusive and
jointly exchaustive possibilities: the continuum hypothesis is provable
in ZFC, it is independent of ZFC, it is refutable in ZFC. Goedel
eliminated the third possibility, on the assumption that ZFC is
consistent, Cohen eliminated the second possibility on the same
assumption. So now we know that either the continuum hypothesis is
independent of ZFC or ZFC is inconsistent. (Or, at any rate, so it can
be proved in Bounded Arithmetic).

I can't think of any possibility about the status of ZFC other than:
it's true (but not provable in ZFC), it's false (but not refutable in
zFC), or it hasn't got a truth-value. I think that covers the
possibilities. If I have it right Feferman thinks that propositions in
the first-order language of arithmetic always have a truth-value, but
propositions in higher-order languages don't necessarily (unless,
perhaps, they can be decided in some predicatively acceptable theory).
The continuum hypothesis is a proposition in the third-order language
of arithmetic. 

I suppose you could draw a distinction between those who find
set-theoretic statements completely meaningless, and those who say they
do have a truth-value but it depends on the model, you have to specify
the model and we haven't currently specified the class of models we are
considering well enough to decide the continuum hypothesis. In this way
the truth-value of the continuum hypothesis depends on matters of
convention which might change in the future so that the continuum
hypothesis gains a truth-value. I myself don't think the latter stance
is very coherent, because you're assuming the metalanguage in which the
model theory is done is meaningful, and I don't see how this can be
without an intended model available.

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