Editing Fixed-point combinator (section)

===Uses===
Applied to a function with one variable, the ''Y'' combinator usually does not terminate. More interesting results are obtained by applying the ''Y'' combinator to functions of two or more variables. The added variables may be used as a counter, or index. The resulting function behaves like a ''while'' or a ''for'' loop in an imperative language.

Used in this way, the ''Y'' combinator implements simple [[recursion (computer science)|recursion]]. The lambda calculus does not allow a function to appear as a term in its own definition as is possible in many [[programming language]]s, but a function can be passed as an argument to a higher-order function that applies it in a recursive manner.

The ''Y'' combinator may also be used in implementing [[Curry's paradox]]. The heart of Curry's paradox is that untyped lambda calculus is unsound as a deductive system, and the ''Y'' combinator demonstrates this by allowing an anonymous expression to represent zero, or even many values. This is inconsistent in mathematical logic.