Thread: Compactness Theorem and partial orderings

1. Compactness Theorem and partial orderings

Hello folks,

I've been asked 'If P is a partial ordering, how do I use the compactness theorem to show that P is the union of k chains iff each finite subset on P is the union of k chains?'

But, I have absolutely no idea what this question is even driving at. Set theory and logic is easily my weakest area of maths and any help would be much appreciated (I will attempt to reciprocate in differenctial equations or algebra - things I can actually do!)

There is a question what a subset of P is. Let's say P is an order on a set A, i.e., $P\subseteq A\times A$. Then a subset of P could mean a subrelation on the same set A. However, if A is infinite, the premise of the statement is trivially false and so the statement is trivially true. Indeed, each finite subrelation has infinitely many isolated points, and each of them is a degenerate chain. So I think the problem is talking about finite subsets of A and restrictions of P on these subsets.