[FOM] non-extensional models of set theory

Ali Enayat ali.enayat at gmail.com
Fri Apr 3 07:01:50 EDT 2015

This is a response to a query of Paul B. Levy (April 1, 2015) who has asked:

>>Is there a result in the literature along the following lines?

>>Given a model of ZF without Extensionality, its "extensional quotient" is a model of ZF including Extensionality.

The answer is in the positive, provided one uses the theory ZF*
instead of ZF, where the axioms of ZF* are the same as ZF except that
in ZF* the collection scheme and the separation scheme take the place
of the replacement scheme of ZF.

Of course ZF and ZF* have the same deductive closure, but it has been
known since Scott's 1961-paper [1] that ZF\{Extensionality} is
interpretable in Zermelo set theory (and therefore its consistency
strength is strictly lower than that of ZF).

I know of two papers in which the details of the interpretability of
ZF in ZF*\{Extensionality} are established by the method of
"extensional quotients"; namely Friedman's [2] and Hinnion's [3]
below. Indeed [2] shows that ZF (with classical logic) is
interpretable in the intuisionistic version of ZF*\{Extensionality}.

Gandy's paper [4], mentioned in Tait's reply (April 1, 2015) to Levy's
query uses a different method to interpret GB (Gödel-Bernays theory of
classes) in GB\{Extensionality}.

[1] D. Scott, More on the axiom of extensionality, in Essays on the
foundations of mathematics,  pp. 115–13, Magnes Press, Hebrew Univ.,

[2] H. Friedman,  The consistency of classical set theory relative to
a set theory with intuitionistic logic, J. Symb. Log. 38 (1973), pp.

[3] R. Hinnion, Extensionality in Zermelo-Fraenkel set theory.
Zeitschr. Math. Logik und Grundlagen Math. 32 (1986), pp. 51–60.

[4] R. Gandy, On the axiom of extensionality. I. J. Symb. Logic 21
(1956), pp. 36–48

The Consistency of Classical Set Theory Relative to a Set Theory with
Intuitionistic Logic Author(s): Harvey Friedman Source: The Journal of
Symbolic Logic, Vol. 38, No. 2 (Jun., 1973), pp. 315-319

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