Dilworth's Decomposition Theorem in the Infinite Case

Now-a-days we should probably say that the infinite case of Dilworth’s decomposition theorem [6] follows from the finite case by “a standard compactness argument.” Depending upon one’s upbringing, what we would have in mind is an application of Godel’s compactness theorem in logic (cf. Church [5]), Tychonoff’s theorem on compact topological spaces [25], or (if one is a combinatorialist) Rado’s selection lemma [19]. For example, we might argue as follows. Let P = 〈P, ≤〉 be a (partially) ordered set with no antichain of size k + 1. Then, by the finite case of Dilworth’s theorem, any finite sub collection of the set of sentences $$S = \{ p_{x1} \,V_{px2} V \cdots Vp_{xk} :x \in P\} \cup \{ \neg \left( {p_{xi} \Lambda p_{yi} } \right):1 \leqslant i \leqslant k,x \bot y\}$$ has a model (where x ⊥ y means x and y incomparable in P, and p xi is a propositional variable with the intended interpretation “x belongs to the i th chain”). Hence S has a model and P is the union of the k chains C i = {x ∈ P: p xi is true (1 ≤ i ≤ k).