Is This the Toughest Math Riddle Ever?

[Jrthedawg]

Give this riddle a read. As an avid fan of math riddles and brain teasers, this may be the most difficult one I've ever encountered.

I posted this in the Set Theory forum, but thought this audience might enjoy it as well. Let me know what you think! Be aware that the solution is at the bottom of the page provided in the link.

 

[mfb]

I saw a similar puzzle with functions R->R a while ago. It is really surprising, but it needs the axiom of choice[/color] (mark to see) - and that is known for the strange result it produces.