Thread: Set-theory challenge

This is a question I was asked on my oral exam for set-theory. I found it a particular interesting one:

Show that:

For any infinite set , there exists a bijection well-orderings-theorem of Zermelo.


It's in particular the implication that's interesting.

Hint 1.

Spoiler:

For : Make a smart choice for

Addition hint 1.

Spoiler:

Let an arbitrary infinite set. Choose:

Originally Posted by Dinkydoe

This is a question I was asked on my oral exam for set-theory. I found it a particular interesting one:

Show that:

For any infinite set , there exists a bijection well-orderings-theorem of Zermelo.


It's in particular the implication that's interesting.

Hint 1.

Spoiler:

For : Make a smart choice for

Addition hint 1.

Spoiler:

Let an arbitrary infinite set. Choose:

Instead of well-ordering can we use Zorn's or AOC?

Well in fact that's allowd too! If you show:

for all infinite there exists a bijection AOC, Lemma of Zorn, Wellorderings Theorem of Zermelo.

They're all equivalent ofcourse.