# Why do we assume there are ordinary and extraordinary sets?

if $R$ were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if $R$ were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that R is neither normal nor abnormal

We have to assume that there are such things as extraordinary or "abnormal" sets for this paradox to be valid. The solution to the paradox is to change the definition of a set so that it cannot include self-referent collections.

Why do we assume that sets must be normal or abnormal in the first place? To my eye this whole thing can be avoided if we do away with what appears to be an unnecessary assumption.

• I have never seen that statement in my life. If you're quoting somewhere, please include a source. – Asaf Karagila Mar 20 '18 at 17:10
• Right ... that's exactly Russell's point: Russell argued that if we are not careful in our definitions, we get a contradiction. And indeed one way to resolve the paradox is to disallow 'abnormal' sets, which was Russell's suggestion. So I think you're on the same page here. – Bram28 Mar 20 '18 at 17:12
• @AsafKaragila the quote is from wikipideia but it is pretty much identical to an explanation of Russels paradox in a Graph Theory book I am reading (Introduction To Graph theory by Richard J. Trudeau ISBN-10: 0486678709) – user543846 Mar 20 '18 at 17:21
• I can live with awful explanations written on Wikipedia. After all, not everyone there is a logician or even a trained mathematician. But in an actual mathematical book? Tsk tsk tsk. – Asaf Karagila Mar 20 '18 at 17:22
• its a good book! at least it is getting me interested in Graph theory, I just am taking issue with this little quibble which is probably in there to make the subject more approachable – user543846 Mar 20 '18 at 17:31

This is the source of the paradox. If the set $R$ is normal, then it has to be abnormal; and if it is abnormal, then it will have to be normal. Since a set is never both normal and abnormal, it can only be that $R$ is not a set to begin with.