 \(\omega\)-consistency and 1-consistency are purely syntactic notions. If the use of the notions of truth and falsity is allowed, the assumption of 1-consistency can be expressed intuitively simply as the requirement that the formal system in question does not prove any false \(\Sigma^{0}_1\)-sentences (i.e., the system is sound at least in the case of such sentences). From now on, it is assumed that the formalized systems under consideration contain Q and are assumed to be at least 1-consistent, unless otherwise stated. 