Re: Cantor's circular "proof" that evens = integers

On May 15, 2:23 pm, george <gree...@xxxxxxxxxx> wrote:
On May 15, 1:35 am, herbzet <herb...@xxxxxxxxx> wrote:
I was aware that "ZFC is consistent" was not provable in
ZFC, by Godel's second incompleteness theorem, unless
ZFC is inconsistent. I thought that if ZFC is consistent,
then it is just possible that ZFC proves "ZFC is inconsistent"
as is hypothesized in AK's question to me.

This is an odd kind of question to be saying "just possible"
about. It is in fact not possible.

But the upshot of its unprovability is precisely that we don't know
this. For all we know, ZFC is consistent and nonetheless proves "ZFC
is inconsistent".

If it were "possible" that
ZFC proved this, then it would also have to be "possible"
(by first-order completeness) for "ZFC is inconsistent" to
be true IN EVERY model of ZFC.

Surely correct.

THAT canNOT be possible:
you could not DEFEND the ALLEGATION that con(ZFC)
ENCODES "ZFC is consistent" if con(ZFC) has the WRONG
truth-value in EVERY model of ZFC!

Interesting argument, but the conclusion doesn't follow. If ZFC is
consistent but proves ~con(ZFC), it only follows that ZFC proves
something false about itself. Indeed, that we quite naturally say
that that is what follows implicitly acknowledges that con(ZFC) is in
fact an adequate representation of ZFC's consistency.

.

Relevant Pages