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AN EXTENSION OF A THEOREM OF ZERMELO

Published online by Cambridge University Press:  06 March 2019

JOUKO VÄÄNÄNEN
JOUKO VÄÄNÄNEN*
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS PO BOX 64 (GUSTAF HÄLLSTRÖMIN KATU 2) FI-00014 UNIVERSITY OF HELSINKI, FINLAND and INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM SCIENCE PARK107 1098XG AMSTERDAM, NETHERLANDSE-mail: jouko.vaananen@helsinki.fi
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Abstract

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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].

Keywords

03E7003C35categoricityZermelosecond-order logicset theory
Type
Communications
Information
Bulletin of Symbolic Logic , Volume 25 , Issue 2 , June 2019 , pp. 208 - 212
Copyright
Copyright © The Association for Symbolic Logic 2019 

References

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