List of ordinals: Difference between revisions
Jump to navigation
Jump to search
no edit summary
No edit summary Tags: Mobile edit Mobile web edit |
No edit summary |
||
Line 82:
* <!--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>
<!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>-->
* Least stable ordinal<ref name=":0" /><sup>(p.6)</sup><ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref><sup>(p.9)</sup>, this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" /><sup>p.23</sup>
* Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" />
| |||