The Y-combinator (or any other fixed-point combinator) turns the lambda calculus into a Turing-complete language by showing that it can express recursion. The l

The Y-combinator (or any other fixed-point combinator) turns the lambda calculus into a Turing-complete language by showing that it can express recursion. The lambda calculus in itself is an extremely simple declarative language. Its simplicity and Turing-completeness makes it a perfect target for programming language research. I think it's fair to say that most modern ideas of programming languages have been demonstrated as an extension of the lambda calculus first. So without the Y-combinator, you might not have Rust's traits or Python's yield, or C#'s LINQ.