[FOM] CH and mathematics

Bill Taylor W.Taylor at math.canterbury.ac.nz
Thu Jan 17 23:43:04 EST 2008

Alex Blum wrote:

> Solomon Feferman asks:
>        "Is the Continuum Hypothesis a definite mathematical problem?"

A good question, deserving a serious attempt at answering.

> Can someone elaborate on what the choices are or could have been?

No-one seems to have tackled this yet,  perhaps becuase it is more a question
in math philosophy rather than foundations; but I will just add a little
comment of my own, which is nowhere near a real answer.

Last year, while list member Thomas Forster was visiting here, a topic
came up in conversation between us, wherein I was querying whether some
mathematical structure "really existed".  His quick reply was, "Well,
we know it's consistent, therefore it exists.  Consistency is existence!"

I apologise to Thomas if I've misrepresented him, but that was the gist
of what he said, I think.  I came back with a response that, surely this
couldn't be the case, because ZFC (for example) could be extended in various
ways, *using the same language with the same intended interpretation*,
(an important point, I think, which I didn't stress at the time),
all of which might well be consistent separately, but not together.

I chose an example which relates to the current thread inquiry; 
whereby CH and ~CH both lead to the existence of certain *objects*,
which CANNOT both exist together, not if sets are "real things".


(1) ~CH implies there exists an infinite subset A of the reals R,
        which surjects neither to Z nor to R.

(2)  CH implies that there exists a function f from P(R), whose value
        at input X, f(X), is a surjection from X to R, or to N,
        or to some finite subset of N.

Loosely speaking, A is a set witnessing that CH is false;
and f is a function that picks out witnesses that it is true.

Now, on a Platonist/realist view of math, one of A or f must exist;
the consistency of of the existence of either with ZFC is known, 
yet the consistency of the existence of both with ZFC is impossible.

a: IMHO, this must destroy Thomas' view that "consistency is existence",
   at least for the mathematical realist.   Do others agree?

b: Also, the answer to which one exists seems to be closely involved with
   Alex's question.  Is this so?

Bill Taylor.

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