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:
Thanks to Mees de Vries.