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diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex index e6377a0..43dc6d0 100644 --- a/ordinal-zoo.tex +++ b/ordinal-zoo.tex @@ -129,35 +129,43 @@ the Veblen functions with up to that many variables. is the proof-theoretic ordinal of Kripke-Platek set theory ($\mathsf{KP}$). -\ordinal The collapse of $\Omega_\omega$ (“Takeuti-Feferman-Buchholz -ordinal”), which is the proof-theoretic ordinal of -$\Pi^1_1$-comprehension. [\CHECKTHIS: there may be a confusion - between $\Omega_\omega$ and $\Omega_{\omega+1}$ in the collapse.] +\ordinal The countable collapse of $\varepsilon_{\Omega_\omega + 1}$ +(“Takeuti-Feferman-Buchholz ordinal”), which is the proof-theoretic +ordinal of $\Pi^1_1$-comprehension + transfinite induction. -\ordinal\label{CollapseInaccessible} The collapse of an inaccessible +\ordinal\label{CollapseInaccessible} The countable collapse of +$\varepsilon_{I+1}$ where $I$ is the first inaccessible (= $\Pi^1_0$-indescribable) cardinal. This is the proof-theoretic ordinal of Kripke-Platek set theory augmented by the recursive inaccessibility of the class of ordinals ($\mathsf{KPi}$), or, on the -arithmetical side, of $\Delta^1_2$-comprehension. -See \cite{JaegerPohlers1983}. +arithmetical side, of $\Delta^1_2$-comprehension + transfinite +induction. See \cite{JaegerPohlers1983}. (Compare •\ref{RecursivelyInaccessible}.) -\ordinal\label{CollapseMahlo} The collapse of a Mahlo cardinal. This -is the proof-theoretic ordinal of $\mathsf{KPM}$. +\ordinal\label{CollapseMahlo} The countable collapse of +$\varepsilon_{M+1}$ where $M$ is the first Mahlo cardinal. This is +the proof-theoretic ordinal of $\mathsf{KPM}$. See \cite{Rathjen1990}. (Compare •\ref{RecursivelyMahlo}.) -\ordinal\label{CollapseWeaklyCompact} The collapse of a weakly compact +\ordinal\label{CollapseWeaklyCompact} The countable collapse of +$\varepsilon_{K+1}$ where $K$ is the first weakly compact (= $\Pi^1_1$-indescribable) cardinal. This is the proof-theoretic ordinal of $\mathsf{KP} + \Pi_3\hyphen\mathsf{Ref}$. See \cite{Rathjen1994}. (Compare •\ref{RecursivelyWeaklyCompact}.) -\ordinal\label{CollapsePiTwoZeroIndescribable} The collapse of a +\ordinal\label{CollapsePiTwoZeroIndescribable} The countable collapse +of $\varepsilon_{\Xi+1}$ where $\Xi$ is the first $\Pi^2_0$-indescribable cardinal. This is the proof-theoretic ordinal of $\mathsf{KP} + \Pi_\omega\hyphen\mathsf{Ref}$. -See \cite[part I]{Stegert2010}. (Compare •\ref{WeaklyStable}.) +See \cite[part I]{Stegert2010} (in whose notation this ordinal would +be called $\Psi^{\varepsilon_{\Xi+1}}_{\mathbb{X}}$ where $\mathbb{X} += (\omega^+; \mathsf{P}_0; \epsilon; \epsilon; 0)$). +(Compare •\ref{WeaklyStable}.) \ordinal The proof-theoretic ordinal of $\mathsf{Stability}$: -see \cite[part II]{Stegert2010}. +see \cite[part II]{Stegert2010} (in whose notation this ordinal would +be called $\Psi^{\varepsilon_{\Upsilon+1}}_{\mathbb{X}}$ where +$\mathbb{X} = (\omega^+; \mathsf{P}_0; \epsilon; \epsilon; 0)$). % \section{Recursively large countable ordinals} |