Ah, OK, I think I now understand your intended construction. I’m trying to figure out whether it satisfies the ZF axioms, but right now it’s past my bedtime. One thing that definitely isn’t true is the following stronger version of “if Foundation holds in the original model then it also holds in your universe”: Write x->y to mean that set y in your universe is obtained from set x in some other model of ZF by “replacing some empty sets with things from a,b,...”). Then (I repeat: this is a thing that isn’t true): If x has an element w disjoint from x, and x->y, and when x->y w turns into z, then z is disjoint from y. So if Foundation is true in your universe it’s not for the very most obvious reason. (Counterexample to that stronger claim: let x = {{{}}, {{},t}} for some choice of t, let w = {{},t}; let y = {{b}, {a,t}} so that w → {a,t}; then although w is disjoint from x it isn’t true that z is disjoint from y, because both contain a={b}.
I’ll return to this tomorrow, if I find the time, and think some more about whether your universe is guaranteed to be a model of ZF...
Ah, OK, I think I now understand your intended construction. I’m trying to figure out whether it satisfies the ZF axioms, but right now it’s past my bedtime. One thing that definitely isn’t true is the following stronger version of “if Foundation holds in the original model then it also holds in your universe”: Write x->y to mean that set y in your universe is obtained from set x in some other model of ZF by “replacing some empty sets with things from a,b,...”). Then (I repeat: this is a thing that isn’t true): If x has an element w disjoint from x, and x->y, and when x->y w turns into z, then z is disjoint from y. So if Foundation is true in your universe it’s not for the very most obvious reason. (Counterexample to that stronger claim: let x = {{{}}, {{},t}} for some choice of t, let w = {{},t}; let y = {{b}, {a,t}} so that w → {a,t}; then although w is disjoint from x it isn’t true that z is disjoint from y, because both contain a={b}.
I’ll return to this tomorrow, if I find the time, and think some more about whether your universe is guaranteed to be a model of ZF...