Why the fourth case only with the Axiom of Choice?

[namespace1]

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,

 

[Valkarie]

The fourth case is not provable without the Axiom of Choice (or an equivalent form) because it is impossible to prove that there does not exist a One-to-One Onto function on A onto a subset of B. The Axiom of Choice provides a method for constructing such a function, and so it is necessary in order to prove that none exists.To prove the fourth case using Zorn's Lemma, you would need to assume the existence of a partial ordering on the set of all possible functions from A to B, such that every chain of functions has an upper bound. Then you could prove that this partial ordering has a maximal element, thus showing that there exists no One-to-One Onto function from A to a subset of B.