I thought I understood chains and antichains in posets, but I'm a little lost with Kunen's definition in Set Theory: An Introduction to Independence Proofs. He says:
"2.2. DEFINITION. Let $\langle \mathbb{P},\leq \rangle$ be a partial order. A chain in $\mathbb{P}$ is a set $C \subset \mathbb{P}$ such that $\forall p,q \in C(p \leq q \cup q \leq p)$."
I'll pause here to note that I suspect the $\cup$ must be a misprint for $\lor$. First I'm having trouble seeing how the union symbol in this property can even be grammatical. The best I can come up with for interpreting it would be akin to saying $(p \leq q \land q \leq p)$ which seems entirely too restrictive to be a useful definition of a chain in anything like the sense it usually means. However, since the rest of the definition loses me, I suspect my confusion stems from this. Anyway, it continues:
"$p$ and $q$ are compatible iff $$ \exists r \in \mathbb{P} (r \leq p \land r \leq q); $$ they are incompatible $(p \perp q)$ iff $\lnot \exists r \in \mathbb{P} (r \leq p \land r \leq q)$. An antichain in $\mathbb{P}$ is a subset $A \subset \mathbb{P}$ such that $\forall p,q \in A (p \neq q \implies p \perp q)$."
I'm confused about why the defintion of antichain seems to be dependent on minimal or even minimum elements. For example, take the set $\mathcal{P} \{ a,b,c \}$ with $\leq$ defined as the usual subset relation. An example of an antichain would be the set $\{ \{ a,b \} , \{ c \} \}$, at least by my usual understanding. But since I can find an $r$ that is $\leq$ both $\{ a,b \}$ and $\{ c \}$, namely $\emptyset$, wouldn't they be compatible by Kunen's definition, and thus not part of an antichain? How could any set with a minimum element ever have antichains?
Yes, $\cup$ is a misprint for $\lor$. This is not the usual notion of antichain in a partial order, but it is the one needed for Martin’s axiom and forcing.
In $\wp(\{a,b,c\})$ no two elements are incompatible, because $\varnothing\le X$ for all $X\subseteq\{a,b,c\}$. The more useful example is $\wp(S)\setminus\{\varnothing\}$ for an arbitrary set $S$, as in Ken’s second example: then incompatible sets are precisely disjoint sets, and antichains are precisely pairwise disjoint families.