Comments by "EebstertheGreat" (@EebstertheGreat) on "The Foundation of Mathematics - Numberphile" video.

@tylerfusco7495 Not every symbol is a set, but in ZFC, every term will include a variable name, and all variables are sets. Same with any other set theory that doesn't have proper classes or urelements.
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It can tell you the proof is valid. It just can't tell you if the proof is invalid. Similarly, I can tell you whenever a given TM halts. Just run it until it does. I simply can't (usually) tell you if the TM fails to halt.
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@joshuahillerup4290 OK. I just didn't quite understand your post in the right way then.
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@tylerfusco7495 Well, in ZFC, every term is either equal to T, F, or a set. So I still think my point holds. There exist sets without terms defining them, but not the other way around. Cause think about it. Of course logic is more general than that. It could apply to anything, not just to sets. But it's logic. Obviously if I claimed that some logical claim was fundamental to any theory of mathematics, you could argue for some particular interpretation of math instead to defeat me. You cold create a sufficiently pathological model in any case unless I added stricter conditions.
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@kazedcat In a theory like NBG, a set is just a sort of class. In NBG, you can prove that every set is a class and also that there exist classes which are not sets (called "proper classes"), such as the class of all sets. No comparable object exists in ZFC. [NF goes much further, despite in some sense being a weaker theory, where basically everything is a set (yes, the "set of all sets" really is a set in NF).]
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