Iterated failures of choice

Abstract

We make an important observation regarding iterated symmetric extensions which allows us to easily iterate certain counterexamples to the axiom of choice. We use this to improve previous results and to obtain a model where: (1) Every partially ordered set can be embedded into the cardinals; and(2) for every non-empty set X there is a set A which is the countable union of countable sets, and P(A) has a Dedekind-finite subset that can be mapped onto X. We also provide an outline for a construction of a model where every field admits a non-trivial vector space whose endomorphisms are all scalar multiplications. This method can be applied to similar "∀∃-∃∀ problems".