I'm looking for different criteria of identity for the notion of 'set'. I know that the standard criterion of identity is extensionality but I was wondering if there are others. I looked around but couldn't find any literature on other criteria. I thought you could help me find some material on that topic?
Thanks for taking the time. All the best,
David
Extensionality is an axiom in ZF set theory; the '=' relation is built-in, i.e., part of the basic apparatus of predicate logic.
(Usually. If you treat '=' as a defined symbol via Extensionality, then you need to explicitly postuate the substitution properties of it.)
However, you said "criterion", so I guess that amounts to the same thing.
You could reverse the direction of the $\in$ relation: $$ x=y \Leftrightarrow \forall z(x\in z \Leftrightarrow y\in z)$$ In ZF, this is a theorem: the Pair axiom gives us the singletons $\{x\}$ and $\{y\}$, and one can show that $y\in \{x\}\Leftrightarrow y=x$. From this it's a short step to this "reversed extensionality".
The founders of axiomatic set regarded extensionality as basic to the very concept of a set. It comes from the notion of "extension of a property", and is the key feature that distinguishes "property" from "set" (or collection or class).
If you seach for "extensionality" in the Stanford Encyclopedia of Philosophy, you will find a number of articles that may bear on your question, depending on how it's interpreted.