  Second, we should note that, although this approach requires the Skolemite to start with an independent argument against our ordinary conception of sets, it need not render the Löwenheim-Skolem theorems themselves completely superfluous. After all, it's still a theorem that set-theoretic notions like countability and uncountability come out relative on the algebraic conception. This isn't something which happens to all set-theoretic notions—e.g., “x is the empty set” or “x has seventeen members”—and it's not something which just drops out from the algebraic conception of axiomatization.