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 $\alpha_n$?