Abstract
In [3], 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.
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