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 $\omega$-compelte poset $X$, and any $\omega$-continuous function $T:X\rightarrow X$, 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?