Re: Request for Peer Review - Refutation of Cantor Theorem Conclusion
- From: "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx>
- Date: 28 Apr 2006 15:09:35 -0700
Patricia Shanahan wrote:
Arturo Magidin wrote:
In article <1146239905.551887.306000@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,...
Scott <ToaTerra@xxxxxxxxx> wrote:
Can you explain how the diagonal avoids being in the power set? Do you
regard it as not being a set, or do you think it contains some element
that is not in the base set?
If D is an element of the powerset, then the powerset/set-theory is
inconsistency. (Its a proof by contradiction.)
That's a tall order. Since the fact that D is an element of the power
set is a trivial consequence of the axioms of ZF, you are saying much
more than just that you have disproven the conclusion of Cantor's
Theorem: you are asserting that you have proven that Zermelo-Fraenkel
Set Theory is inconsistent!
Looking over some of his messages, I think Scott may be starting out by
assuming there is a surjection from N onto P(N). Cantor's Theorem does
prove an inconsistency in a set theory that adds that, as an additional
axiom, to ZF.
Patricia
If the mapping between x and f(x) is f(x) = x+1, with ordinals, the
unmapped element is the empty set. With f(x) = x, the unmapped
element, for x E N, is N. Where N E N, nothing is missing.
One way to show that Cantor's powerset result and part of the
"uncountability of the reals" does not have the traditional or standard
interpretation is to illustrate that the theory and other theories that
would support said standard results may not be consistently applied,
for example where there is a universe and there is no universe in ZF
set theory, and coincidentally there is a universe and it is its own
powerset.
That is all quite explained in some hundreds and thousands of posts
here on sci.logic and sci.math.
It's nice to consider where there are only "countably" many reals
between, say, zero and one. For example, in the classical
infinitesimal analysis, it is assumed to be that way, and in
delta-epsilonics, es muss sein. Heh. I read a book about probability
and it is expressed in terms of game theory instead of measure theory,
and they seem quite happy to be rid of measure theory and its noisome
discontent. In more about the countability of the reals, basically
that gets into nilpotent/differential versus Newton/Robinson
infinitesimals.
Scott, I think here earlier you were trying to tell people that only
true statements could be encoded. Then, we critiqued your paper, about
Goedel. Post a draft of your paper here and it will be criticized,
constructively.
There's only one theory with no axioms. There's a universe, and
there's no universe in ZF, so, where there is and isn't and that
violates mutual exclusivity, or LEM, and that's bad, ZF is
inconsistent.
There is no universe in ZF.
Ross
.
- References:
- Request for Peer Review - Refutation of Cantor Theorem Conclusion
- From: Scott
- Re: Request for Peer Review - Refutation of Cantor Theorem Conclusion
- From: Scott
- Re: Request for Peer Review - Refutation of Cantor Theorem Conclusion
- From: David C . Ullrich
- Re: Request for Peer Review - Refutation of Cantor Theorem Conclusion
- From: Scott
- Re: Request for Peer Review - Refutation of Cantor Theorem Conclusion
- From: Arturo Magidin
- Re: Request for Peer Review - Refutation of Cantor Theorem Conclusion
- From: Patricia Shanahan
- Request for Peer Review - Refutation of Cantor Theorem Conclusion
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