# Axiom of Extensionality defined in the other direction?

Why we cannot define axiom as two sets are equal iff they are element of same sets? What are the problems with that approach?

It doesn't seem like I quantify over anything rather than sets. The definition may be trivial if we assume that every set define a set which have only one element and it is that set. In that case definition will be trivial and every set will be different but let assume that these are the sets we are usually working with (ZFC). However intutitively, I can provide you every different set, but still I can define a set which do not bear that property. You may define a new set to show my set is different than others but it seems like you changed the set, by definition I defined that set as being part of these sets but not the sets that make it different than all other sets.

My reasoning may not clear but if you can ask me to read some stuff related to these, I will be grateful.

My idea is denying the classical axiom of extensionality and pair axiom I guess and replace extensionality definition with some kind of reverse of classical axiom of extensionality. ( I am hoping that the classical one will not follow from converse if I deny pair axiom and introduce undecidability in terms of elements of sets.)

My idea probably something formalized like that:

∀x∀y(∀z(x∈z↔y∈z)→x=y)

Thanks to Mees de Vries.

• Last sentence on this section. – Git Gud Sep 24 '16 at 16:57
• What if I want to deny the classical axiom of extensionality and just accept that converse ? – user371928 Sep 24 '16 at 16:59
• @GitGud, I feel like the question is about a "reverse" extensionality axiom where two sets are postulated to be equal given that they are members of the same sets, rather than that their members are the same sets. That is, something like $\forall x \forall y (\forall z(x \in z \leftrightarrow y \in z) \to x = y)$. – Mees de Vries Sep 24 '16 at 17:09
• You can look at whatever axioms for set theory you want. The only question is why - what does this kind of axiom system do that you like, besides be different from ZFC? – Noah Schweber Sep 24 '16 at 17:10
• ''reverse'' is what I mean, I just realized converse so it didnt strike me as the thing that I do not mean in the first place, I can use your formulation in the question if you allow me. – user371928 Sep 24 '16 at 17:11