List of ordinals: Difference between revisions
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** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal
** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability
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* The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma\)<ref name="Welch2010Draft" />
* Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" />
* The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref>
*<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. -->
* Least start of a gap in the constructible universe of length 2<ref name="Gaps" />
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* Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models />
* The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" /><sup>(p.8)</sup>
The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists".
== Uncountable ordinals ==
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