Also, and a big also, the Y-combinator exists in (untyped) lambda calculus---e.g. all you need is abstraction and application. This came as a shock to Church w

Also, and a big also, the Y-combinator exists in (untyped) lambda calculus---e.g. all you need is abstraction and application. This came as a shock to Church when he invented it as he wanted to use LC as a language for mathematical logic and fixed point combinators spell out doom for logical purposes. He thus invented the simply typed lambda calculus to banish such constructions.