User:Emlightened/Type Theories

Currently working on making blog posts for introducing type theories; they're underrated in googology imo.

[User:Emlightened/Type_Theories/I] Is Gödel's \(\mathcal T\), which has PTO \(\varepsilon_0\).

Intend to add:

ML<ω (PTO \(\Gamma_0\))

Extension of ML2(?) with impredicative polymorphic eliminator for naturals (PTO unknown)

ML<ωW (PTO \(\sup_{n<\omega}\psi_Omega{I_n}\))

Extension of ML1 with \(\mu\) types with impredicative inductive polymorphic eliminator (PTO unknown)

Spektor's \(\mathcal B\) (equiconsistent with analysis)

System Fω (equiconsistent with higher order arithmetic)

Calculus of Inductive Constructions (equiconsistent with ZFC)

Here impredicative polymorphism (where explicitly mentioned) is used for eliminators which can produce terms of types outside the universe they belong to. For instance, this would let us create a function from the naturals to the church numerals for a type operator.

ML will be formalised with types \(0, 1, 2, \mathbb N\) and type constructors \(\Sigma, \Pi\), alongside appropiate universe types.