 \(\aleph_0\) is the cardinality of the set of natural numbers, and \(\aleph_1\) is the next transfinite  cardinal.[9]  When we supplement classical quantificational logic with \(Q_0\), read: “at least \(\aleph_0\) objects”, we must relinquish Compactness even if we can still retain the Löwenheim-Skolem theorem. The opposite is the case, however, when we supplement classical quantificational logic with \(Q_1\), read: “at least \(\aleph_1\) objects” instead.