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?
What does Tarski's Fixed-Point theorem give us that that Y-Combinator does't
lambda calculus;recursion;type theory;partial order;lattices
If by Tarski's fix point theorem you mean the KnasterTarski fixpoint theorem, then it's widely applicable and very general. All you need is a complete lattice and a monotone function on the lattice.  There are many rather different examples of those. Tarski-Knaster is for example used in the coinductive definition of bisimilarity. Another application is to construct models of logical formulae (e.g. for logics with fixpoints, such asthe modal $\mu$-calculus and the various fixpoint logics that are used to characterise complexity classes). It is also used in set theory and in program analysis.The Y-combinator on the other hand is a specific piece of syntax in the $\lambda$-calculus. You don't need to prove the existence of the Y-combinator.