Re: the problem with Cantor

On Aug 7, 1:23 pm, Aatu Koskensilta <aatu.koskensi...@xxxxxx> wrote:

Study of ZFC + CH might well be sensible for any number of
reasons. ZFC + ~CH on the other hand is an utterly pointless theory.

Oh, come ON. Surely the same was _originally_ "felt", if not overtly
expressed, about non-standard models of PA, which were arguably
discovered with G1 around 1930. But after 1960, it turned out that
non-standard arithmetic could justify infinitesimals, which *was* at
least _A_ point. Seriously, given that you have said this, you
deserve to live to see somebody actually come up with some respectable
application for models of ZFC + ~CH. I'm not alleging that your
expectation that this will not happen is under-informed, rather, I'm
just saying that IN LIGHT OF PAST DEVELOPMENTS, it seems, well,
hubristic.


Usually, behind claims such as that you refer to is the idea that
since the continuum hypothess is undecidable in ZFC, its truth or
falsity is an indeterminate matter, and that mathematics bifurcates,
so to speak, into mathematics with CH and without CH, according to the
whim and fancy of the mathematician. What is mysterious is that no
reason is given why we shouldn't similarly regard mathematics as
bifurcating into mathematics with "ZFC is inconsistent" and
mathematics with "ZFC is consistent" given that "ZFC is inconsistent"
is just as undecidable in ZFC as CH.

This is ridiculous; reasons for not doing this ARE SO TOO given, the
loudest being that if "ZFC is inconsistent" is actually true, then
models of
ZFC+"ZFC is inconsistent" CANNOT EXIST AT ALL.
If ZFC is inconsistent then it has NO models, So howEVER you
bifurcate, you obviously canNOT bifurcate THAT way (this IS a reason
and it IS the reason GIVEN).
If you do, you have a statement with the property that you are
alleging that it means "ZFC is inconsistent", yet you have a model of
ZFC, in which this statement is true.
The model is OBVIOUSLY *wrong* about this, since if it were right,
it(the model) COULD NOT EVEN EXIST in the first place! If you want to
study models of Th+~Con(Th), for ANY Th, you almost have to DENY that
Con(.)MEANS "consistent"! THAT IS NOT THE CASE for denying AC or CH.

It remains of course a striking empirical fact that no such
"bifurcation" is in reality to be found in mathematics. It is merely a
formalist pipe-dream of no actual substance.

--
Aatu Koskensilta (aatu.koskensi...@xxxxxx)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Also, if
any two relatively consistent extensions of ZFC are equally valid we
are led to the strange conclusion that "ZFC is inconsistent" is just
as valid as "every set is constructible". Neither of these are
regarded as having much "validity" by most set theorists
.

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