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List of ordinals: Difference between revisions

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List of ordinals (edit)

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(Seems unusual to see "L_(recursive ordinal) n P(omega) models (a theory of second-order arithmetic)")
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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
* The least ordinal in \(E_\eta\), for \(\eta > 0E_1\).<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> In unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\).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]'' (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 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" />
* Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" />
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