Earlier quoted context omitted.
> a Subset cannot have greater cardinality than the set it derives from. I am not completely certain in the subtleties of non-standard set theory, but at least if we replace the axiom of Choice (AC) by the axiom of Determinacy (AD) - which is perfectly fine, though it is less common to do mathematics in this system - we can partition the real line R by an equivalence relation that has more equivalence classes than el…
> we can partition the real line R by an equivalence relation that has more equivalence classes than elements of R That's not surprising, since an equivalence relation is a function of two elements of the set, i.e. O(N^2), and a subset is a function of a single element, i.e. O(N) . The OP mentioned subsets, and an equivalence relation is not a subset of the original set--it doesn't even typecheck.
I don't understand your reasoning. Every equivalence relation partitions the set S on which it is defined on into subsets - these are called equivalence classes. Every equivalence class contains at least one element of S. So I find it quite surprising that there are more equivalence classes than elements of S (in this case S are the real numbers) if we assume AD.