and does this mean there cant be sets of things I have not defined equivalency for?
This is maybe a weird question and maybe not even a meaningful one, my understanding of sets and such is not complete, but it seems like "two sets are the same if they contain the same elements" implies that for all elements in both sets I can decide if they are equivalent or not. Specifically I thought of this question because I think in computer science I came across sets of Turing machines for which equivalency in general can not be determined because it would solve the halting problem (if equivalency is defined as same input leads to same output considering halting and not halting). Is such a set then ill-defined and not actually a set?
Equivalence relations are mainly used to partition a given set, not normally a relation between sets.