Is Russell's Paradox Truly Unsolvable in Traditional Logic?

[kuengb]

I got this little thing second-hand from a computer science student. One of his professors mentioned it in the Logic lecture. Define the set M as follows:
$M:=\{x \mid x \notin x\}$

This brings up a strange contradictory since
$M\in M \Rightarrow M\notin M$
and
$M\notin M \Rightarrow M\in M$

As my information is correct there was a big discussion among mathematicians when these lines were written down the first time since it somehow contradicts the logic axiom that something is either true or false. Is that true (or false )? Does anyone know something about this?

 

[matt grime]

it's just russell's paradox that states naive set theory is not the thing you want to use. some things are too big to be sets, or if you like, just saying a set is a collection of objects with a rule for belonging or not belonging is not sufficient. see zermelo frankel set theory aka ZF