AN EXTENSION OF A THEOREM OF ZERMELO

We show that if (M, ∈1, ∈2) satisfies the first-order Zermelo–Fraenkel axioms of set theory when the membership relation is ∈1 and also when the membership relation is ∈2, and in both cases the formulas are allowed to contain both ∈1 and ∈2, then (M, ∈1) ≅ (M, ∈2), and the isomorphism is definable i...

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Veröffentlicht in:	The bulletin of symbolic logic 2019-06, Vol.25 (2), p.208-212
1. Verfasser:	VÄÄNÄNEN, JOUKO
Format:	Artikel
Sprache:	eng
Schlagworte:	Axioms Categoricity Communications First order theories Hypotheses Isomorphism Logical theorems Mathematical logic Mathematical relations Mathematical set theory Mathematical theorems Set theory Symmetry Theorems Zermelo Frankel set theory
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Beschreibung
Zusammenfassung:	We show that if (M, ∈1, ∈2) satisfies the first-order Zermelo–Fraenkel axioms of set theory when the membership relation is ∈1 and also when the membership relation is ∈2, and in both cases the formulas are allowed to contain both ∈1 and ∈2, then (M, ∈1) ≅ (M, ∈2), and the isomorphism is definable in (M, ∈1, ∈2). This extends Zermelo's 1930 theorem in [6].
ISSN:	1079-8986 1943-5894
DOI:	10.1017/bsl.2019.15