I’ve run into a problem in philosophy recently that I do not completely appreciate. Certain sets are said to be “too big” to be sets. In Lewis’ Modal Realism, the set of all possible worlds is said to be one such set. These are sets whose memberships is composed of infinite individuals of a robust cardinality. I (purportedly) understand that not all infinities are equal. But I don’t quite see why there can be a set of continuum many objects, but not a set of certain larger infinities. Am I misunderstanding what it is to have “too big” a set?
As Alex says, in Lewis's case, he's really pointing towards an idea familiar from the philosophy of set theory. Not all "collections" of objects can form sets: The assumption that they do leads to contradiction. (Of course, we need some logical assumptions to get that contradiction, and these could be denied. And one might also not think contradictions are all that bad. But let's not go there now.) And given the standard axioms of set-theory, we can prove that there is no set containing every object and, moreover, that there is no set that can be put in 1-1 correspondence with all the objects there are. But the standard concept of set, as embodied in Zermelo-Fraenkel set-theory, is not in any way motivated by the idea that a set cannot be "too big". It is based upon a very different idea, called the "iterative"conception of set, though there is some question about whether the Axiom ofReplacement (which was Fraenkel's distinctive contribution) is really motivated by the iterative conception or...
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