List of ordinals: Difference between revisions

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* The least (next recursively inaccessible ordinal)-stable ordinal

* The least (next recursively Mahlo ordinal)-stable ordinal

* The least (next \( \Pi_n \)-reflecting ordinal)-stable ~~ordnal~~, for \( 2<n<\omega \)

* The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \)

* The least doubly \( (+1) \)-stable ordinal<ref name=":0" />(p.4)

* The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal

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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 \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" />

* 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 \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref>

* 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>

*

* Least start of a gap in the constructible universe of length 2<ref name="Gaps" />

* Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" />

* Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" />

* Least \( \beta \) that starts gap of length \( \beta^+ \) = ~~least~~ ~~height~~ of model of KP+"\( \omega_1 \) exists".<ref name=":0" />(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref>

* Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref>

* Least start of third-order gap = least ~~height~~ of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />(p.6)

* Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />(p.6)

*

* Least height of model of ZFC<ref name=":0" />(p.6)

@@ Line 57: / Line 57: @@
 * The least (next recursively inaccessible ordinal)-stable ordinal
 * The least (next recursively Mahlo ordinal)-stable ordinal
-* The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \)
+* The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \)
 * The least doubly \( (+1) \)-stable ordinal<ref name=":0" /><sup>(p.4)</sup>
 * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal
@@ Line 72: / Line 72: @@
 * 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" />
-* 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 \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref>
+* 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>
 *<!-- 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" />
 * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" />
 * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" />
-* Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" /><sup>(p.6)</sup> <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref>
+* Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" /><sup>(p.6)</sup> <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref>
-* Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" /><sup>(p.6)</sup>
+* Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" /><sup>(p.6)</sup>
 * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer-->
 * Least height of model of ZFC<ref name=":0" /><sup>(p.6)</sup>

List of ordinals: Difference between revisions

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