Recursion in Calculus

Assume you want to write a recursive function that returns the factorial of a natural number. You might attempt something like:
F = Ln.(((If-then-else)(Zero?)n)1)((*)n)(F)(Pred)n

This, however, does not work as explicit self-references are not permitted.

Fortunately, the problem can be solved using the so-called fixed point operator (also called paradoxical operator or Y combinator):
Y = Ly. (Lx.(y)(x)x) Lx.(y)(x)x

Applying Y on a simple example:

> (Y)f (L y. (L x. (y) (x) x) L x. (y) (x) x) f ---> (L x. (f) (x) x) L x. (f) (x) x ---> (f) (L x. (f) (x) x) L x. (f) (x) x ---> (f) (Y)f

Note that the use of Y would immediately end up in an endless recursion in case of right-most order evaluation. Scheme permits self-referential definitions instead.

	Assume you want to write a recursive function that returns the factorial of a natural number. You might attempt something like: `F = Ln.(((If-then-else)(Zero?)n)1)((*)n)(F)(Pred)n`
	This, however, does not work as explicit self-references are not permitted.
	Fortunately, the problem can be solved using the so-called fixed point operator (also called paradoxical operator or Y combinator): `Y = Ly. (Lx.(y)(x)x) Lx.(y)(x)x`
	Applying `Y` on a simple example: