Are there any areas of mathematics that are known to be impossible to formalise in terms of set theory?
Are there any areas of mathematics that are known to be impossible to formalise in terms of set theory?
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set-theory
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0Which set theory? One cannot really formalize Category Theory in ZFC, because one cannot talk about, e.g., the category of all groups in ZFC. – 2011-02-25
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0Set theory in general? If category theory cannot be formalised in ZFC, is there are another system in which it can be formalised? – 2011-02-25
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5There is no "set theory in general", that's the point. But if you add certain axioms about large cardinals, then you can formalize category theory; even without them, you can formalize what is "almost as good" as category theory, namely small category theory (where the collection of all objects is a set, and the collection of all arrows between two objects is a set). – 2011-02-25
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5One can use NBG when dealing with categories. I'm sure one of out resident logicians will shortly come to the rescue.... – 2011-02-25
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6With some seriousness, I would give the following answer: "Yes -- set theory." – 2011-02-25
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1@Pete: Well, I would point out that, in fact, Goedel used set theory to formalize set theory (in the sense of constructing an "inner model" for set theory within set theory, V=L...) though that could be somewhat of a stretch, perhaps. – 2011-02-26
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0Isn't one of the ZFC axioms actually an axiom schema that generates an axiom when given a set? You can't universally quantify that "axiom" since you'd have to start with "for any set S in the set of all sets", which doesn't exist, no? – 2011-03-13
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1Most set-theorists regard the treatment of universes in category theory as rather clumsy in comparison with the far richer and subtle theory of large cardinals in set theory, of which it is a very small part. And to my knowledge, all the answers to foundational questions about the strength of the various universe axioms in category theory have come from set theory. – 2011-06-13
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2There is no technical reason that prevents someone from using ZFC as a metatheory for studying ZFC. The "metaverse axioms" of Joel Hamkins, Victoria Gitman, and their colleagues can be viewed as something of this sort, set theory studied within set theory. – 2011-06-14
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0Thanks, Carl! (Here is the paper to which I think you refer: http://arxiv.org/abs/1104.4450) – 2011-06-14
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0@Carl: in my old comment above I didn't mean to suggest that it was somehow forbidden or strictly impossible to study set theory using set theory. That would be a strangely counterfactual statement, since after all contemporary set theorists do this. My point was rather that this is an incredibly complicated and delicate game: we do not know that the ZFC axioms are consistent, they are certainly incomplete, and so forth. So it seems that set theory has not been (completely) formalized, at least not yet. – 2011-06-14