User blog comment:Ubersketch/Weird axiomatic system/@comment-35470197-20190606034604

Good. If ZFC is inconsistent in the base theory, the resulting formal theory is just ZFC set theory (formalised in ZFC set thoery itself). On the other hand,, if ZFC is consistent in the base theory, then the resulting formal theory is a maximal consistent extension of ZFC set theory. Here, the maximality means that every statement is either provable or disprovable. We usually construct such a formal theory using Zorn's lemma (which is equivalent with the axiom of choice), but your construction actually works in ZF set thoery if I am correct.