One might wonder why, when moving from the standard world to the non-standard world, elements of sets become "*-elements" or "pseudo-elements", but equality remains equality. This is because equality is not a predicate! Rather, it serves to identify objects precisely (for instance, any model guarantees that the formula "x=A" (where A is some constant and x a variable) is true for precisely one x). As such, it has a specific interpretation in the model.

This technical issue has various consequences. But since anything provable in the non-standard world is also provable in the standard one, clearly there's no particular cause for concern.