\(UT(2^{\aleph_{0}},2^{\aleph_{0}},2^{\aleph_{0}})\): If every member of an infinite  set of cardinality \(2^{\aleph _{0}}\) has power \(2^{\aleph_{0}}\), then the union has power \(2^{\aleph_{0}}\).

\(UT(\aleph_{0},2^{\aleph_{0}},2^{\aleph_{0}})\): The union of denumerably many sets each of power \(2^{\aleph _{0}}\) has power \(2^{\aleph_{0}}\).

There is an ordinal \(\alpha\) such that for all \(X\), if \(X\) is a denumerable union of denumerable sets then \({\cal P}(X)\) cannot be partitioned into \(\aleph_{\alpha}\) non-empty sets.