In , Takeuti developed the theory of ordinal numbers (ON) and constructed a model of Zermelo-Fraenkel set theory (ZF), using the primitive recursive relation ∈ of ordinal numbers. He proved:
(1) If A is a ZF-provable formula, then its interpretation A0 in ON is ON-provable;
(2) Let B be a sentence of ordinal number theory. Then B is a theorem of ON if and only if the natural translation B* of B in set theory is a theorem of ZF;
(3) (V = L)° holds in ON.