Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.



Re: Cantor's absurdity, once again, why not?
Posted:
Mar 15, 2013 3:41 AM


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 ZFaxiom of extensionality requires a technique to identify (= distinguish from others) every element of a set. For nonmaterial 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 wellordered 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



