# [FOM] The independence of Extensionality

Ali Enayat enayat at american.edu
Tue Apr 26 23:49:45 EDT 2005

This is a reply to a query of Thomas Forster, who has asked (April 26,
2005):

" Who was it who first proved the [independence of] axiom of extensionality
from the other
axioms of ZF?  I'm thinking in particular of tricks like: all sets are
empty (in the new sense)  unless they are singletons and $x \in \{y\}$
(in the new sense) if $x \in y$ (in the old).   Was it Fraenkel and
Mostowski? And can anyone supply a reference..?"

To my knowledge the first paper that discusses the intricacies of the axiom
of extensionality, in the framework of first order logic, is:

Abraham Robinson, On the independence of the Axioms of Definiteness (Axiome
Der Bestimmtheit)
The Journal of Symbolic Logic, volume 42 (June 1939), pp. 69--72.

Two other landmark papers are the following (I learnt about this whole topic
through Scott's paper below. It disabused me of the notion that the
extensionality axiom is a "trivial" axiom of ZF).

Robin Gandy, On the axiom of extensionality. I.
The Journal of Symbolic Logic, volume 21 (1956), pp. 36--48,

and

Dana Scott, More on the axiom of extensionality.
Essays on the foundations of mathematics pp. 115--131 Magnes Press, Hebrew
Univ., Jerusalem, 1961

Best regards,

Ali Enayat



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