Countability: Difference between revisions

<nowiki>In terms of ordinals, it is clear that \( \omega \) is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable infinite ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using successors and countable unions. In particular, it is an </nowiki>[[Epsilon numbers|epsilon number]]<nowiki>, and much more. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki>