Successor ordinal

An ordinal is called a successor if it is equal to \(\alpha + 1\) for some other \(\alpha\). An ordinal \(\alpha\) is a successor if and only if \(\max \alpha\) exists. The smallest successor ordinal is 1, which is also the only successor ordinal to also be additively principal. The least ordinal that is not a successor, other than 0, is \(\omega\).

If \(\beta\) is successor, then \(\alpha+\beta\) is also successor for all \(\alpha\). However, multiplication and exponentiation do not have this property.