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Re: Cantor's absurdity, once again, why not?
Posted:
Mar 15, 2013 3:41 AM
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On 15 Mrz., 02:51, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> I'll pass on discussing this with you. I've had my fill of debating
> anything with you for a while.
Your decision. Quite understandable.
Nevertheless readers should know that the ZF-axiom of extensionality
requires a technique to identify (= distinguish from others) every
element of a set. For non-material elements this means labelling (by
names, words, definitions). Since it is impossible for all elements of
an uncountable set we have a contradiction.
In principle AC is quite right. Only its application to uncountable
sets is impossible. There it has the same logical status as the
following axioms:
1) There are numbers that cannot be identified.
2) There are values that have not values.
3) There is a set of n positive natural numbers with sum n^2/2 (AMS).
4) There are two prime triples.
5) There is a smallest positive real number (not identifyable, of
course). This can be extended to:
6) There is a permutation of all rational numbers such that they are
well-ordered by magnitude.
... and many, many more of this sort, all based upon the basic idea
that some numbers cannot be identified or that there is a finished
unfinished.
Regards, WM
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