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AN EXTENSION OF A THEOREM OF ZERMELO
Published online by Cambridge University Press: 06 March 2019
Abstract
We show that if $(M,{ \in _1},{ \in _2})$ satisfies the first-order Zermelo–Fraenkel axioms of set theory when the membership relation is
${ \in _1}$ and also when the membership relation is
${ \in _2}$, and in both cases the formulas are allowed to contain both
${ \in _1}$ and
${ \in _2}$, then
$\left( {M, \in _1 } \right) \cong \left( {M, \in _2 } \right)$, and the isomorphism is definable in
$(M,{ \in _1},{ \in _2})$. This extends Zermelo’s 1930 theorem in [6].
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- Copyright © The Association for Symbolic Logic 2019
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