The Axiom of Choice constrains every set of cardinal numbers which is linearly ordered by size to be well-ordered. By contrast, does ZF-without the Axiom of Choice (but with the Axiom of Foundation)-allow complete liberty in the ordering of cardinal numbers by size? More precisely: Is the following statement consistent with ZF? "If S is any linearly ordered set, there exists a set of cardinal numbers which is linearly ordered according to size and which is ordinally similar to S". Even without the Axiom of Foundation one can define a partial ordering among the sets of ZF. The set A precedes the set B if there exists an injective mapping of A into B. The set A strictly precedes the set B if there exists an injective mapping of A into B but no mapping of A onto B. One could then ask the same question about the linearly ordered subsets of this partially ordered class of sets.

tentativeanswer, is yes, this is consistent. $\endgroup$ – Asaf Karagila Aug 21 '14 at 19:05