Embedding higher kinded types (monads!) into the untyped lambda calculus

Question

It\'s possible to encode various types in the untyped lambda calculus through higher order functions.

Examples:
zero  = λfx.      x
one   = λfx.     fx
two   = λ         


        

Answer 1

You are mixing up the type level with the value level. In untyped lambda calculus there are no monads. There can be monadic operations (value level), but not monads (type level). The operations themselves can be the same, though, so you don't lose any expressive power. So the question itself doesn't really make sense.