[FOM] AC & CH (was counted sets)

Bill Taylor W.Taylor at math.canterbury.ac.nz
Mon Aug 24 00:52:26 EDT 2009

Before the thread on AC and CH disappears completely, I might
observe one trivial little result which is not widely known
or remarked on, showing that these two are perhaps somewhat
intimately related.  Especially if one has high regard
for the as-yet-informal concept of "explicit" existence,
as the "counted sets" thread indicates some people might.

As was observed here, most mathematicians are happy enough with
the use of countable/dependent choice, as the objects thereby
deemed to exist can (in almost every case) be explicitly produced.
On the other hand there is more uneasiness with the use of full
choice, for example to produce non-measurable sets of reals,
free ultrafilters, and similar matters.

If one followed a math philosophy of "explicit existence only",
one might happily accept ZF, (allowing that power sets were
of explicit subsets only - still an undefined concept), but
demur at ZFC.  One would then have a kind of resolution of CH.

Taking the view that (in the enforced absence of a proof either way)
sets did not exist unless explicitly definable in some sense,
one would then note that CH was both true AND false!  That is,
the statement of ZFC could be re-worded in two different ways,
that were equivalent in ZFC but not in ZF.

Specifically, (1) if one requires that there exist a set of
cardinality greater than N but less than R, such a set could
not be produced, so CH would be true.

On the other hand, (2) one might require that a function exist
from P(R) to P(R)^P(R) , such that if   N subset X subset R,
then f(X) is either a bijection from N to X, or from X to R.
That is, given X, there is a "canonical" witness to either
the countability or continuumhood of X.  In this latter case,
no such function exists, so CH would be deemed to be false.

To show the equivalence (in ZFC) of these two is a simple matter;
and I would hazard a guess that most concerned mathematicians
would favour the former rather than the latter, as being
the "proper" interpretation of CH in ZF. 

But it is intriguing that the dichotomy exists.

-- Bill Taylor

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