Additive principal ordinals: Difference between revisions
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An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least |
An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least additive principal ordinal is 1 since \(0 + 0 < 1\), and all additive principal ordinals other than 1 are limit ordinals. In particular, as can be seen from the Cantor normal form theorem (every ordinal has a [[Cantor normal form|CNF]] representation), additive principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some ordinal \(\gamma\). As such, the second infinite additive principal ordinal is [[Omega^2|\(\omega^2\)]], the first limit of additive principal ordinals is [[Omega^omega|\(\omega^\omega\)]], and the first limit of the limits of additive principal ordinals is \(\omega^{\omega^2}\). |
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Additive principal ordinals can be generalized to multiplicative principal ordinals and exponential principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicative principal ordinals are to additive principal ordinals as additive principal ordinals are to limit ordinals. However, exponential principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just \(\omega\) and the [[epsilon numbers]]. |
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Latest revision as of 16:53, 25 March 2024
An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least additive principal ordinal is 1 since \(0 + 0 < 1\), and all additive principal ordinals other than 1 are limit ordinals. In particular, as can be seen from the Cantor normal form theorem (every ordinal has a CNF representation), additive principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some ordinal \(\gamma\). As such, the second infinite additive principal ordinal is \(\omega^2\), the first limit of additive principal ordinals is \(\omega^\omega\), and the first limit of the limits of additive principal ordinals is \(\omega^{\omega^2}\).
Additive principal ordinals can be generalized to multiplicative principal ordinals and exponential principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicative principal ordinals are to additive principal ordinals as additive principal ordinals are to limit ordinals. However, exponential principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just \(\omega\) and the epsilon numbers.