Generalizations of Cantor's theorem in

A set x is Dedekind infinite if there is an injection from ω into x ; otherwise x is Dedekind finite. A set x is power Dedekind infinite if , the power set of x , is Dedekind infinite; otherwise x is power Dedekind finite. For a set x , let pdfin(x ) be the set of all power Dedekind finite subsets of x . In this paper, we prove in (without the axiom of choice) two generalizations of Cantor's theorem (i.e., the statement that for all sets x , there are no injections from into x ): The first one is that for all power Dedekind infinite sets x , there are no Dedekind finite to one maps from into pdfin(x ). The second one is that for all sets , if x is infinite and there is a power Dedekind finite to one map from y into x , then there are no surjections from y onto . We also obtain some related results.