namespace1
Jan14-11, 01:41 PM
Hello people:
IN cardinal domination, a representative set A is dominated by a representative set B iff there's a One-to-One Onto function on A onto a subset of B. This yields four logical cases as the following:
Let X dom Y denote "X is dominated by Y".
- X dom Y & Y dom X (ByCantor/Shroder/Brenstien Theorem card X equals card Y)
- X dom Y & not(Y dom X)
- not(X dom Y) & Y dom X
- not(X dom Y) & not(Y dom X) ---> My problem is here
Why can't we prove the fourth case without using the Axiom of Choice or an equivalent form of it? A technical answer would be great as trying to prove it using Zorn's Lemma.
This is not a homework question. I'm a Math enthusiast and a self-learner who works on Set Theory. :)
Thanks in advance,
IN cardinal domination, a representative set A is dominated by a representative set B iff there's a One-to-One Onto function on A onto a subset of B. This yields four logical cases as the following:
Let X dom Y denote "X is dominated by Y".
- X dom Y & Y dom X (ByCantor/Shroder/Brenstien Theorem card X equals card Y)
- X dom Y & not(Y dom X)
- not(X dom Y) & Y dom X
- not(X dom Y) & not(Y dom X) ---> My problem is here
Why can't we prove the fourth case without using the Axiom of Choice or an equivalent form of it? A technical answer would be great as trying to prove it using Zorn's Lemma.
This is not a homework question. I'm a Math enthusiast and a self-learner who works on Set Theory. :)
Thanks in advance,