Re: Request for Peer Review - Refutation of Cantor Theorem Conclusion

In article <1146267438.998710.5030@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Scott <ToaTerra@xxxxxxxxx> wrote:
Magidin:
you are asserting that you have proven that Zermelo-Fraenkel
Set Theory is inconsistent!

Okay, seems extreme to me, but you're the expert.

I'm not an expert in set theory.

I'm inclined more
towards the idea that the predicate used in D is invalid because it
attempts to return both true and false, or neither.

I simply do not agree that is the case. Please see my quoting of the
direct proof elsewhere in this thread.

Magidin:
"s is not in f(s)" is well-defined

But if "s is not in f(s)" returns both true and false, is it still
well-defined?

Well, if my grandmother had wheels, she'd be a bicycle. But then, she
doesn't have wheels.

S is a ->given<- set. We already know it is a set. f is a ->given<-
function from S to P(S). We already know it is a function. That means
that for every s in S, f(s) ->must be<- a subset of S.

Now, given any subset A of S, the predicate "s is not in A" is
perfectly well defined for every element of S. Because A is already
given to be a subset, and each element will either be in A or not be
in A; one, or the other, and not both.

So all we do in D is to pick A to depend on s in a way that we
->already<- have given, namely f.

So what makes you think that the predicate "tries to return both true
and false"?

Perhaps you attempted to argue by contradiction, and assumed that
there existed an s in S such that f(s)=D?

Then why do you believe that the problem is not in the assumption that
such an s exists, rather than in the construction of the predicate?
Imagine we go back and agree that "If my grandmother had wheels, then
she would be a bicycle", and then we agree that the assumption that my
grandmother has wheels leads to a contradicction. Does that suggest to
you that there is a problem with the notion of "bicycle"?

Can I not remove the contradiction by claiming D is an indirect
instance of the class of sets {x:x notin x} and therefore does not
exist?

You can make a lot of claims; unfortunately this is unfounded, because
D is defined explicitly as a subset of S. It is defined to be the
collection of all s IN S with a given property; and the Axiom of
Specification guarantees that such a thing is a set (and a subset of
S).

In general, how would you recommend resolving this problem? You have
one proof that shows D exists and (assume for the sake of discussion)
another proof showing D does not exist? Perhaps this will highlight
some area I am not familiar with.

You go over them and try to find the mistake; either one of them is
wrong, or else the axiom system is inconsistent.


--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

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