If you look at the set of finite subsets of $\mathbb N$ (side question: Is there a standard notation for that?) partially ordered by the subset relation, you see that it can be partitioned into maximal antichains (that is, you have a set of maximal antichains so that any two of them are disjunct, and their union is the set of finite subsets of $\mathbb N). One possibility (but by far not the only one) is to have each antichain consist of all sets of the same cardinality.
Also for the set of all subsets of $\mathbb N$ that are either finite or cofinite, it's not hard to define such a partition; partition the finite sets according to their cardinality, and the cofinite sets according to the cardinality of the complement.
Of course such simple strategies won't work any more with the full $\mathcal P(\mathbb N)$. Indeed, I have no idea how one would either define such a partition, or prove that there doesn't exist one. Of course there's also the possibility that there exists such a partition, but you cannot explicitly specify it.
Therefore my question is: Does there exist a partition of $(\mathcal P(\mathbb N),\subseteq)$ into maximal antichains, and if so, is is possible to explicitly specify one?