I\'m taking a graduate course on the theory of functional programming, based on

Question

I'm taking a graduate course on the theory of functional programming, based on Paul Taylor's "Practical Foundations of Mathematics." I understand the statement of Tarski's theorem about how for any ?-compelte poset X, and any ?-continuous function T:X?X, that T has a fixed point which is the join of (The statement and proof can be found here).

What I want to know is, how is this applicable, other than being a proof that the Y-combinator exists? It just seems to me that it says "we can use recursion to build a function that is defined for all natural numbers", where we could use some other recursive type for numbers. Doesn't the existence of the Y-combinator show the same thing?

Explanation / Answer

If by Tarski's fix point theorem you mean the Knaster

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