Emulating recursion, and more generally the whole Church-Turing thesis. The fact that you can represent mechanical computation through, well, computation - the fact that equating those concepts makes sense at all, which was such a huge innovation that today it seems too banal to even notice.
You keep answering with the same words though. My question was _where_ the Y combinator is useful, not _what_ it is designed for, which I am of course aware of.
What do you mean "where"? In Princeton? In the Proceedings of the London Mathematical Society? In the head of anyone who presumes to think about computation?
