Re: Request for Peer Review - Refutation of Cantor Theorem Conclusion
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Fri, 28 Apr 2006 16:14:33 +0000 (UTC)
In article <1146239905.551887.306000@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Scott <ToaTerra@xxxxxxxxx> wrote:
Hi All:
Thanks for the feedback.
You should attribute the quotes you are giving. You have taken replies
from many posts and written them as if they were all written by the
same person.
Okay. First, IF you could show that the diagonal set is not an element
of the power set, this would NOT establish that "f is [...] a
bijection." The proof begins with an arbitrary function f:S->P(S), so
at best it would establish that f ->could be<- a bijection.
Assume the following:
1. D is not an element of the powerset.
2. A mapping function f:N->P(N).
What other things are necessary to show f is a bijection between N and
P(N)?
Depends on what you mean by "mapping function". I understand that to
mean simply that it is ->a<- function; meaning that it satisfies the
following:
(i) f is a subset of N x P(N).
(ii) For each n in N there exists A in P(N) such that (n,A) is in f
(iii) For all n in N and all A,B in P(N), if (n,A) and (n,B) are in
f, then A=B.
(We usually write f(n)=A for the unique A such that (n,A) is in f).
To make it a bijection, it must also satisfy two ancillary conditions:
(iv) f is one-to-one: for all n,m in N, A in P(N), if (n,A) and
(m,A) are in f, then n=m.
(this would be equivalent to saying that f(n)=f(m) if and only if
n=m).
and
(v) f is onto: for all A in P(N) there exists n in N such that
(n,A) is in f.
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!
If you have some other
collection of axioms for set theory in mind you need to
start by saying exactly what the axioms are.
Nope, standard axioms.
Then why so modest? You have not "merely" disproven Cantor's Theorem,
you have proven the axioms of set theory are 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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