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Re: "not everything in math can be described by sets” This is probably a better example: Russel's Paradox. Does there exist a set of all sets that do not conta

Re: "not everything in math can be described by sets” This is probably a better example: Russel's Paradox. Does there exist a set of all sets that do not contain themselves? I think it's a clear example of something that can't be described by standard set theory. "Set Theory", without further qualification, I think that it's safe to assume that they are talking about ZFC (or maybe Naive) set theory. Also, this is a reply to a post, so I need to make some assumptions about the audience, while keeping the post brief and coherent. I'm not trying to open up a discussion on what statements are true in every possible version of set theory, or the implication in other branches of math.

constantcrying 3y ago
>I think it's a clear example of something that can't be described by standard set theory. It absolutely can be described by naive set theory. You literally just did it. It also is not a mathematical object in ZFC, there is no way to construct it. >I'm not trying to open up a discussion on what statements are true in every possible version of set theory, or the implication in other branches of math. But that is the only way the statement could ever be false. In standard mathematics, founded on ZFC, every object is a set. So the only way the statement could ever be false is if you look at different foundations and broader theories.