Am 03.12.12 15:21, schrieb jaakov: > Given a set X and a cardinal k, is there a set Y such that card(Y)=k and > X is disjoint from Y? > > Is there a proof of this fact that works without the axiom of regularity > (= axiom of foundation) and does not assume purity of sets? >
I am not sure if I understand all of this.
Consider the class of all ordinals not in X. This is a well-ordered proper class and hence has an initial segment which is order isomorphic to k. Take that initial segment.