Additive principal ordinals: Difference between revisions

An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least additivelyadditive principal ordinal is 1 since \(0 + 0 =< 1\), and all additivelyadditive principal ordinals other than 1 are limit ordinals. In particular, youas can seebe seen from the Cantor normal form theorem (every ordinal has a [[Cantor normal form|CNF]] representation), that the additivelyadditive principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some ordinal \(\gamma\). As such, the second infinite additivelyadditive principal ordinal is [[Omega^2|\(\omega^2\)]], the first limit of additivelyadditive principal ordinals is [[Omega^omega|\(\omega^\omega\)]], and the first limit of thosethe limits of additive principal ordinals is \(\omega^{\omega^2}\).

<nowiki>AdditivelyAdditive principal ordinals can be generalized to multiplicativelymultiplicative principal ordinals and exponentiallyexponential principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicativelymultiplicative principal ordinals are to additivelyadditive principal ordinals as additivelyadditive principal ordinals are to limit ordinals. However, exponentiallyexponential principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just \(\omega\) and the </nowiki>[[epsilon numbers]].