# Set-theory challenge

• Jan 28th 2010, 08:55 AM
Dinkydoe
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 $X$, there exists a bijection $f:X\times X \to X \Leftrightarrow$ well-orderings-theorem of Zermelo.

It's in particular the implication $\Longrightarrow$ that's interesting.

Hint 1.
Spoiler:
For $\Longrightarrow$: Make a smart choice for $X$

Spoiler:
Let $A$ an arbitrary infinite set. Choose: $X= A\cup \aleph(A)$
• Jan 28th 2010, 02:45 PM
Drexel28
Quote:

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 $X$, there exists a bijection $f:X\times X \to X \Leftrightarrow$ well-orderings-theorem of Zermelo.

It's in particular the implication $\Longrightarrow$ that's interesting.

Hint 1.
Spoiler:
For $\Longrightarrow$: Make a smart choice for $X$

Let $A$ an arbitrary infinite set. Choose: $X= A\cup \aleph(A)$
for all infinite $X$ there exists a bijection $f: X\times X\to X\Leftrightarrow$ AOC, Lemma of Zorn, Wellorderings Theorem of Zermelo.