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



Aatu Koskensilta wrote:

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.

True, but we are discussing the case of ZFC being consistent and proving
"ZFC is inconsistent".

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.

This sentence seems equivocal. Do you mean

a) "However, it's a rather strange idea to hold that infinite sets are
logically possible entities even if the [ACTUAL] inconsistency of
the theory describing them follows from the basic principles
concerning them."

or

b) "However, it's a rather strange idea to hold that infinite sets are
logically possible entities even if the [FALSE ASSERTION OF THE]
inconsistency of the theory describing them follows from the basic
principles concerning them."

I would agree with (a), but that is not what we are discussing.

With (b) I am currently in doubt as to whether it can possibly
obtain that ZFC is consistent and proves a formula that may be
construed as asserting "ZFC is inconsistent", and what it would
mean if that situation _did_ obtain.

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.

Yes, this would not really be satisfactory.

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.

Yes; when we interpret the little epsilon as "is an element of", the
axioms appear to be true statements about collections of things --
although we had a rude shock once with Russell's paradox.

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

Always a possiblity.

or proves
false arithmetical statements,

I'll take your word for this, since I am struggling with the
concept of a consistent theory proving a "false" proposition.
A consistent theory has models, and what it proves is true in
all of them. If a consistent theory proves a false proposition,
it must be that the structure in which we are interpreting the
theory is not a model of the theory.

both of which would seem like a good reason
to ditch ZFC.

Yes, though what would probably happen in reality would be that
the search would be on for a patch, like what happened after
Russell's paradox was discovered.

The situation you have hypothesized seems quite pathological.
I feel like you're going a long way around to make some point.


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.

It may take me awhile to do that.

--
hz

--
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