 This is where the “largeness” of \(\Omega_n\) comes into play. One (easy) way of guaranteeing this consists in letting \(\Omega_n\) be the \(n^{th}\) uncountable regular cardinal, that is \(\Omega_n\coloneqq \aleph_n\). However, such strong set-theoretic assumptions can be avoided. For instance, it suffices to let \(\Omega_n\) be the \(n^{th}\) recursively regular ordinal (which is a countable ordinal) (see Rathjen 1993a).