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The definition of the Peano numerals has two constructors, Zero: N and Succ: N -> N. Russell's paradox (assuming that we're talking about the idea of "set of a
by andrewla 8y ago
The definition of the Peano numerals has two constructors, Zero: N and Succ: N -> N. Russell's paradox (assuming that we're talking about the idea of "set of all sets that are not members of themselves") is avoided simply because the objects produced are not sets, and sets have no exalted position within the mechanics of category theory.
Talking about concepts like "the category of categories that don't contain themselves" is kind of navel-gazey; and ends up falling apart in most constructive variants just because you can't give a comprehensive construction of elements of this category.
Admittedly I'm throwing some concepts from intuitionalism and type theory in the mix here; if I took the time I could make these statements more precise.
- throwawaymath 8y agoI see what you're getting at. My (similarly hand wavey) perspective is that constructions using category theory end up defining natural numbers as categories of sets with cardinality n. What I was really getting at is, how do you define your relation for the category in such a way that isn't pathological ("n is the category of sets with cardinality n except all such sets containing n")? Alternatively, what are you selecting as the objects for your category if not sets of a given cardinality? To be clear, I consider a lot of discussion about the foundations of set theory (and paradoxes thereof) to be pretty navel-gazey.