Can we partition an infinite cardinal greater than aleph null, into countable number of cofinal subsets? Can we have restriction on the cardinality of cofinal subsets?
For example, suppose $\alpha$ is an infinite cardinal which is the supremum of countably many cardinals $\alpha_n$ strictly less than it. Now, can we have partition into cofinal subsets $V_n$ with cardinality $\alpha_n$?
Yes. Put in set $A_n$ those ordinals of the form $\alpha+n$ where $\alpha$ is $0$ or limit. One can do better, of course, since $\kappa=\kappa\times\kappa$ for any infinite cardinal $\kappa$, so we can in fact find $\kappa$ many cofinal subsets. This is explicit, since Gödel's pairing gives us an explicit bijection between $\kappa$ and $\kappa\times\kappa$.