Question

If a function is surjective it has a right inverse. It seems like this can only be true if we grant ourselves the axiom of choice. Is this correct?

Answer 1

Take a surjective function \(f : X \to Y\). Since it is surjective we know that for any given \(y \in Y\) there exists a non-empty preimage \(\{x; f(x) = y\}\). So to construct a right inverse \(f^{-1} : Y \to X\) such that \(f(f^{-1}(y)) = y\) it would seem that for each of the preimages mentioned previously we would need to choose one element. Thus, we need the axiom of choice. Is that indeed true?

Answer 2

Finally, are there any interesting consequences if we reject the axiom choice and lose the implied existence of right inverses for surjective functions?

Answer 3

@tcarroll010 perhaps?

Answer 4

i don't remember much of my set theory lectures but AC is the centerpiece of ZFC. Mathematics without AC is very different. As u pointed out, there r lots of things that may seems trivial but depend deeply on AC.

Answer 5

u could take a look at:
"the axiom of choice" by Jech
"Discovering Modern Set Theory" 2vols by Just and Weese

Answer 6

I suppose it really takes a book to discuss the consequences of the axiom of choice.In any case, thanks.

Answer 7

There are some very strange results like the Banach-Tarski paradox.

Answer 8

u r welcome. i remember i loved my set theory lectures. some crazy things!