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

On 2007-05-14, in sci.logic, herbzet wrote:

Aatu Koskensilta wrote:

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.

"ZFC is inconsistent" is not logically impossible in the technical sense, of
course. However, it's a rather strange idea to hold that infinite sets are
logically possible entities even if the inconsistency of the theory
describing them follows from the basic principles concerning them. From the
completeness theorem we know that if ZFC is consistent there is an
interpretation of the language of set theory in the naturals under which all
of its axioms come out true. But in saying that infinite sets are logically
possible entities surely this is not what we have in mind, since infinite
sets are not horribly complex configurations of naturals under some
arithmetical relation.

We use set theoretical principles to prove stuff like 'the algorithm T
terminates on every n', among other uses. Since the consistency of ZFC is
insufficient to guarantee the truth of such statements given their
provability in ZFC, it seems our willingness to accept proofs of them in
which set theoretical principles are used indicates we are in fact
implicitly committed to stronger soundness properties than mere consistency.
Suppose one day we discover a proof of 'ZFC is inconsistent' from set
theoretical principles. Surely the sensible reaction would be to give up ZFC
for proving arithmetical statements -- either ZFC is inconsistent or proves
a false arithmetical statements, both of which would seem like a good reason
to ditch ZFC.

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.

It means ZFC proves "there is a statement A of form 'V_omega_omega |= P'
such that ZFC |- A, but in fact it is not the case that 'V_omega_omega |=
P'". This is an ordinary set theoretical statement expressible in the
language of set theory, and you'll find the details of formalizing it in any
number of textbooks.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

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

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