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


Hi, I'm having some trouble with my news-server -- It seems to
be running about 24 hours behind. I'm replying via Googles'
wretched web-interface, but I think I'll wait another day before
making any further replies, to see if the teranews server
straightens itself out.

george wrote:
Aatu Koskensilta wrote:

Before addressing your other comments, let me ask you a question.

herbzet wrote:

Sounds like a trick question coming!

Are
infinite sets logically possible entitites if ZFC is consistent but proves
"ZFC is inconsistent"?
Sure, why not?

Because of Godel's second incompleteness theorem,
THAT'S why not.

If ZFC is consistent but proves "ZFC is inconsistent" then
ZFC proves a false statement.

This is simply not a possible state of affairs.
ZFC *does NOT* prove EITHER of "ZFC is consistent" OR
"ZFC is inconsistent" UNLESS ZFC is in fact inconsistent.
*THAT* IS provable in ZFC.

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.

I couldn't see, off-hand, why that would not be possible
(though it seems extremely unlikely to me).

If it were not, in fact, possible then AK's question would
really be a trick question!

Proving a false statement does not mean
an (interpreted) axiom system is inconsistent, or that the false statement
proven is not logically possible.

It does mean that one or more of the axioms are, under interpretation,
false -- that's all.

Under WHAT interpretation?!?

Under the same interpretation we are applying to the theorem
under consideration. In this particular case, it is the
interpretation
that allows us to read the theorem as "ZFC is inconsistent".

If the statement is not provable then, IN ADDITION to
this interpretation under which it is false, THERE MUST ALSO
be one under which it is true. Therefore, even referring to first-
order sentences (as produced by standard inference from
first-order axiom-sets) as true or false AT ALL is stupid.
Truth-values come FROM MODELS in this paradigm.
If all models (of the axioms) agree, then the statement is not
MERELY true or false but (also) provable or disprovable (from
the axioms), AND SHOULD BE REFERRED TO as such (in
EXTREME *preference* to "true" or "false). If one model or
another is going to decide the question in opposite ways then
each simply gives the lie to any claim that the statement has
a truth-value withOUT reference to a model.

I agree (I think); but _my_ point (such as it is) is that the
axioms imply the theorems in _every_ structure, not
just the structures in which the axioms are true (the
models of the axioms). The truth or falsehood of
the axioms in a given structure have _no bearing_ on
what they do or do not imply.

Also it seems, BTW, that such a proof
would have to be a pure existence
proof, since ZFC, being consistent, could not prove both Phi and ~Phi,

I repeat, if ZFC is consistent, it cannot prove EITHER of
con(ZFC) or ~con(ZFC),

For the moment, I'll leave this to the big dogs to hash out!

so the question of whether it could
prove both is quite moot.

What about if ZFC proves "ZFC proves a false
statement about the omega_omega'th level
of the cumulative hierarchy"?

I don't really know what "false statement" means in this context,

Don't panic; you didn't know what it meant in any other
context either, and neither does anybody else, and neither
do they know what "cumulative hierarchy" means.

Yes. At the moment, I'm talking about logical possiblity
_as_ "mathematical truth"; it may fall out of this discussion
that I learn something more about "mathematical truth" than
I currently know.

--
hz

.

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