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In Bonn we've the discussion about the following topic: Pretty much what the title says: Suppoose that A and B is are classes and that there are injections from A to B and fom B to A. Does it follow that there is a bijection between A and B? Example: Let A the class of sets of cardinality one and let B be the class of sets of cardinality two. There is an injection A -> B sending a to {a, emptyset}, B-> A sending b to {{b}}. Does it follow that there is a bijection between A and B? |
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Ignoring set-theoretic technicalities of formulating the question properly, I see no reason that the usual proof of Schroder-Bernstein wouldn't work. (Set-theoretic technicalities: In the standard language of set theory, you can't quantify over classes, so you can't quite state this. However, you can prove a metatheorem saying that whenever you exhibit two such injections, you can prove there is also a bijection. Alternatively, you could work in set theory with classes, in which the statement can be made properly and you ought to be able to prove it just like ordinary Schroder-Bernstein. Alternatively, it is a trivial corollary of the "global" axiom of choice (which implies, in particular, that all proper classes have the same size), though this is kind of applying a sledgehammer.) |
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To ask this question, you have to first be clear what you mean by a "class". Do you mean a finite formula in ZFC language with one free variable, P(x)? (Writing P(x) roughly means "x has property P".) Second, what do you mean by a map from a class P to a class Q? Do you mean a class of ordered pairs? If "yes" to both of the above, then I think the answer to your question is also "yes", because the Schroeder-Bernstein argument will allow you to explicitly write down a formula F(x,y) from the formulas P( ) and Q( ) that is a "bijection" in the sense that for all x such that P(x) there is a unique y such that Q(y) and F(x,y). If, however, you mean class in some other sense, like the undefined notion of "class" used in NBG set theory, or something more vague, a different answer will be required. I recommend reading the wikipedia article on Zermelo-Frenkel Set Theory and browsing related articles until you feel comfortable the precise meanings of the basic terminology :) |
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