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0 pointsicen6y ago0 comments

Indeed - the Type/Kind hierarchy is unified when you write a dependently typed system. In simply typed lambda calculus, you have a system of values, types, and kinds, which are required to make sense of the structure. In dependent types, you have values, and types, which are also values.

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uryga6y ago

> Type/Kind hierarchy is unified when you write a dependently typed system

i'm being pedantic, but i don't think that's always the case. afaik

  Type : Type

leads to "inconsistency" (sounds like Russell's paradox, but my knowledge of the theory gets flaky here). so e.g. in Coq there's actually a hierarchy of "universes":

> "[The type of Int is Set, ] the type of Set is Type(0), the type of Type(0) is Type(1), the type of Type(1) is Type(2), and so on."

http://adam.chlipala.net/cpdt/html/Universes.html

(though a dependently-typed system may hide universes from the user by default and offer "universe polymorphism" to allow them to gloss over it)

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