One might wonder why, when moving from the standard world to the non-standard world
s 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
=A" (where A is some constant and x
) 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.