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.