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Just out of curiosity (the subject is way out of my league at the moment) I have been reading a little about set theory, and I came across Russel's paradox. From what I understood, Russel's paradox proves that we cannot have a set containing everything (real numbers, complex numbers, chickens, spoons), and that some collections of objects are simply not sets. Are there any simple, layman examples of collections of objects that are not sets?

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  • $\begingroup$ I think proper classes are an example (though there may be some set-theoretic subtleties I'm missing...) $\endgroup$ – MathematicsStudent1122 Apr 30 '16 at 6:37
  • $\begingroup$ Actually, Cantor's proof shows that you can't have a set containing everything. Russel showed that $\{ x \mid x \not\in x \}$ is not a set. $\endgroup$ – Christopher Carl Heckman Apr 30 '16 at 6:46
  • $\begingroup$ Set theory is a mathematical theory, and the objects of the universe of set theory are sets. When "layman objects" become abstract mathematical objects, then you can apply Russell's paradox to them. Until then, you can't. Not everything can be explained in layman terms, with layman examples. Sometimes you have to earn the knowledge and intuition by trudging through the math for a while. There is no way around it. $\endgroup$ – Asaf Karagila Apr 30 '16 at 6:59
  • $\begingroup$ Wikipedia's layman terms explanation using "abnormal sets". Cantor said that sets should be definite, and the collection of "normal sets" is not. $\endgroup$ – Pedro Sánchez Terraf Apr 30 '16 at 12:19
  • $\begingroup$ Collections of objects that are not sets don't seem to come up very much if at all in mainstream mathematics. Yes, I'm sure there are rarefied domains in which they may be indispensable, but you can probably do all of classical analysis without them. $\endgroup$ – Dan Christensen Apr 30 '16 at 21:34

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