What is the relationship of the following to other axioms of $\sf ZFC$?

$\sf WB$: Every set $A$ is in bijection with a set well-founded by $\in$.

Obviously, $\sf ZF$ implies $\sf WB$ (because every set is well-founded) and $\sf ZFC-Reg$ implies $\sf WB$ (because every set is equinumerous to an ordinal, which is well-founded). Is it known if $\sf ZF-Reg$ implies $\sf WB$? Another equivalent statement is:

$\sf WB'$: Every set injects into ${\rm WF}:=\bigcup_\alpha V_\alpha$.

(Since every set well-founded by $\in$ is in $\rm WF$, $\sf WB\to WB'$, and for the converse note that the range $B\subseteq {\rm WF}$ of any injection from set $A$ must also be a set so it is contained in $V_\delta$ where $\delta$ is the supremum of the ranks of elements of $B$.) When written this way it seems hard to believe that it could be false even without Regularity.


1 Answer 1



The whole point of urelements is that you can replace them with Quine atoms. Namely sets of the form $x=\{x\}$. Now when you do a permutations model construction, the well founded sets will continue to satisfy choice, if it was true before. Take any non-well orderable set, it cannot inject into a subclass where every set can be well ordered.


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