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



Aatu Koskensilta wrote:

On 2007-05-09, in sci.logic, herbzet wrote:
Whether "THAT infinite set was PRODUCED BY *the axiom of infinity*!"
or had some existence "prior" to the axioms, my personal opinion
is that, assuming consistency of the axioms of ZFC, that infinite
set is a logically possible entity, whatever its status as an
actually existent entity, or as a conditionally existent entity.

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

Sounds like a trick question coming!

Are
infinite sets logically possible entitites if ZFC is consistent but proves
"ZFC is inconsistent"?

Sure, why not? If ZFC is consistent but proves "ZFC is inconsistent" then
ZFC proves a false statement. 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.

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,
or (Phi & ~Phi). This could be headache-inducing to explain, or explain
away.

What about if ...

That's two questions!

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, so
I can't answer this question.

In the previous question, the term "inconsistent" has a clear
syntactical meaning.

Hope this helps!

--
hz

"Even a blind pig occasionally finds an acorn."

--
Posted via a free Usenet account from http://www.teranews.com

.

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