## § Predicative v/s Impredicative: On Universes in Type Theory

#### § Tarski formulation of univereses

• U is a code for types.
• We have a decoding function T: U → Type.
U type

a ∈ U
-----
T(a) type

• Universes bump up levels when we quantify over them?
• Impredicative universes are closed under quantification over types of that same universe.
Q. Given a definition, give me an algorithm that says when the definition needs predicativity A. perform type + universe inference on the definition. If we get a constraint of the form Type lower = Type higher where lower < higher, then the definition needs impredicativity.