The Wikipedia page on Russell's paradox states

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.

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    $\begingroup$ I have never seen that statement in my life. If you're quoting somewhere, please include a source. $\endgroup$ – Asaf Karagila Mar 20 '18 at 17:10
  • $\begingroup$ 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. $\endgroup$ – Bram28 Mar 20 '18 at 17:12
  • $\begingroup$ @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) $\endgroup$ – user543846 Mar 20 '18 at 17:21
  • $\begingroup$ 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. $\endgroup$ – Asaf Karagila Mar 20 '18 at 17:22
  • $\begingroup$ 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 $\endgroup$ – user543846 Mar 20 '18 at 17:31

Your quote seems to be taken from Wikipedia page Russell's paradox. Specifically from the section titled Informal presentation.

Which is exactly what this is: an informal presentation. Which begins by setting an ad-hoc terminology of what is a normal set, and any set which is not normal will be called abnormal. But it is important to emphasize that this is an ad-hoc terminology for the sake of explanation. This is like me saying "let's call a person who can type single-handedly with their non-dominant hand groovy and otherwise they will be called broody", we have created an artificial property of humans, it's not something intrinsic to being a human being, but every human being is either groovy or broody.

But by the law of excluded middle, given a set, it has to be one of the two: normal, and otherwise abnormal. And this is another important point, given a set, it either satisfies the property of "being normal", or it doesn't, in which case it is "abnormal".

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.

  • $\begingroup$ then with the current definition of a set what would constitute an abnormal set? $\endgroup$ – user543846 Mar 20 '18 at 17:19
  • $\begingroup$ If by "current definition" you mean something like modern approaches to sets à la ZF or ZFC, then all sets are "normal". Let me also emphasize, again, how ad hoc and terrible this terminology is. $\endgroup$ – Asaf Karagila Mar 20 '18 at 17:21
  • $\begingroup$ I am not a mathematician - maybe the terriblness of the terminology of the informal presentation is why I find issue with it. Still, I am not sure what the benefit of stating that there must be normal and abnormal catagories of sets just so we can conclude that it is impossible for there to be abnormal sets. $\endgroup$ – user543846 Mar 20 '18 at 17:28
  • $\begingroup$ 1) a given set has to be normal or abnormal. 2) a set cannot be abnormal ... i dont get it. $\endgroup$ – user543846 Mar 20 '18 at 17:29
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    $\begingroup$ Somewhat importantly, no one in the argument being presented claims that there aren't any abnormal sets, either. The paradox will yield a perfectly good contradiction even if we don't have the the axioms of modern set theory that force all sets to be normal. $\endgroup$ – Henning Makholm Mar 20 '18 at 17:39

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