Successor ordinal: Difference between revisions
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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 [[Additive principal ordinals|additively principal]]. The least ordinal that is not a successor, other than [[0]], is [[Omega|\(\omega\)]]. |
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 [[Additive principal ordinals|additively principal]]. The least ordinal that is not a successor, other than [[0]], is [[Omega|\(\omega\)]]. |
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Latest revision as of 16:43, 25 March 2024
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.