- #1

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You can make an inclusion map from X to P(X) if map each element of X to the singleton set containing it in P(X). Thus you have this chain of maps

[tex]X\subset \mathcal{P}(X) \subset \mathcal{P}(\mathcal{P}(X)) \subset \dotsb \subset \mathcal{P}^n(X)\subset \dotsb[/tex]

is there a limit, in the sense of category theory? Can it simply be the union (of the images under inclusion) of all these sets?

I think probably that construction should exist. The category of sets is complete, so all small limits exist.

On the other hand, if you just think about what the final result of such a process will look like, it doesn't really look like a set. For example, if you start with the empty set, you should get the collection of all possible grammatical pairings of opening and closing braces. I think this collection is not well-founded, i.e. is an element of itself, and is therefore not a set.

So my question is: does this construction work? Is it a limit? And if so, is the result a set?