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Line 69: \( \zeta \), 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 * The least ordinal in \(E_1\).,<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> Inin unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\).<ref>R. S. Lubarsky, "ITTMs with Feedback", in ''[http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf Ways of Proof Theory]'', edited by R. Schindler, Ontos Series in Mathematical Logic (2010?, p.338).</ref> * The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" /> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies ~~arithmetically~~\(\mathsf{AQI}\), arithmetical 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 \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" />{{verification failed}} * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[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 \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref>