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1 results

AN EXTENSION OF A THEOREM OF ZERMELO
JOUKO VÄÄNÄNEN
Journal:

Bulletin of Symbolic Logic / Volume 25 / Issue 2 / June 2019

Published online by Cambridge University Press:

06 March 2019, pp. 208-212

Print publication:

June 2019
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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].