How do I define y-combinator without “let rec”?

Question

In almost all examples, a y-combinator in ML-type languages is written like this:

let rec y f x = f (y f) x
let factorial = y (fun f -> function 0 -> 1         


        

Answer 1

I would say it's impossible, and asked why, I would handwave and invoke the fact that simply typed lambda calculus has the normalization property. In short, all terms of the simply typed lambda calculus terminate (consequently Y can not be defined in the simply typed lambda calculus).

F#'s type system is not exactly the type system of simply typed lambda calculus, but it's close enough. F# without let rec comes really close to the simply typed lambda calculus -- and, to reiterate, in that language you cannot define a term that does not terminate, and that excludes defining Y too.

In other words, in F#, "let rec" needs to be a language primitive at the very least because even if you were able to define it from the other primitives, you would not be able to type this definition. Having it as a primitive allows you, among other things, to give a special type to that primitive.

EDIT: kvb shows in his answer that type definitions (one of the features absent from the simply typed lambda-calculus but present in let-rec-less F#) allow to get some sort of recursion. Very clever.