Generalizations of Cantor's theorem in

Abstract

AbstractA 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.