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| -rw-r--r-- | ordinal-zoo.tex | 70 |
1 files changed, 37 insertions, 33 deletions
diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex index 9a25dc3..2fef5e0 100644 --- a/ordinal-zoo.tex +++ b/ordinal-zoo.tex @@ -28,6 +28,7 @@ \newcommand{\REFTHIS}{\textcolor{brown}{REF THIS}} \newcommand{\CHECKTHIS}{\textcolor{orange}{CHECK THIS}} \newcommand{\FIXTHIS}{\textcolor{orange}{FIX THIS}} +\newcommand{\FINDTHIS}{\textcolor{orange}{FIND THIS}} % \newtheorem{ordinalcnt}{Anything}[section] %\newcounter{ordinalcnt}[section] @@ -123,20 +124,21 @@ 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}. +(Compare •\ref{RecursivelyInaccessible}.) \ordinal\label{CollapseMahlo} The collapse of a Mahlo cardinal. This is the proof-theoretic ordinal of $\mathsf{KPM}$. -See \cite{Rathjen1990}. +See \cite{Rathjen1990}. (Compare •\ref{RecursivelyMahlo}.) \ordinal\label{CollapseWeaklyCompact} The collapse of a weakly compact (= $\Pi^1_1$-indescribable) cardinal. This is the proof-theoretic ordinal of $\mathsf{KP} + \Pi_3-\mathsf{Ref}$. -See \cite{Rathjen1994}. +See \cite{Rathjen1994}. (Compare •\ref{RecursivelyWeaklyCompact}.) \ordinal\label{CollapsePiTwoZeroIndescribable} The collapse of a $\Pi^2_0$-indescribable cardinal. This is the proof-theoretic ordinal of $\mathsf{KP} + \Pi_\omega-\mathsf{Ref}$. -See \cite[part I]{Stegert2010}. +See \cite[part I]{Stegert2010}. (Compare •\ref{WeaklyStable}.) \ordinal The proof-theoretic ordinal of $\mathsf{Stability}$: see \cite[part II]{Stegert2010}. @@ -160,32 +162,31 @@ admissibles. This ordinal is not admissible. This is the smallest $\alpha$ such that $L_\alpha \cap \mathscr{P}(\omega)$ is a model of $\Pi^1_1$-comprehension. -\ordinal The smallest recursively inaccessible ordinal: this is the -smallest ordinal which is admissible and limit of admissibles. This -is the smallest ordinal $\alpha$ such that $L_\alpha \models -\mathsf{KPi}$, or, on the arithmetical side, such that $L_\alpha \cap -\mathscr{P}(\omega)$ is a model of $\Delta^1_2$-comprehension. -(Compare •\ref{CollapseInaccessible}.) - -This is the -smallest ordinal $\omega_1^{\mathsf{E}_1}$ not the order type of a -well-ordering recursive in the Tugué functional $\mathsf{E}_1$ -(\cite[chapter VIII, theorem 6.6 on p. 421]{Hinman1978}), or -equivalently, recursive in the hyperjump; and for this $\alpha$ the -$\alpha$-recursive (resp. $\alpha$-semi-recursive) subsets of $\omega$ -are exactly the the subsets recursive (resp. semi-recursive) in -$\mathsf{E}_1$ (\cite[chapter VIII, corollary 4.16 on - p. 412]{Hinman1978}). +\ordinal\label{RecursivelyInaccessible} The smallest recursively +inaccessible ordinal: this is the smallest ordinal which is admissible +and limit of admissibles. This is the smallest ordinal $\alpha$ such +that $L_\alpha \models \mathsf{KPi}$, or, on the arithmetical side, +such that $L_\alpha \cap \mathscr{P}(\omega)$ is a model of +$\Delta^1_2$-comprehension. (Compare •\ref{CollapseInaccessible}.) + +This is the smallest ordinal $\omega_1^{\mathsf{E}_1}$ not the order +type of a well-ordering recursive in the Tugué +functional $\mathsf{E}_1$ (\cite[chapter VIII, theorem 6.6 on + p. 421]{Hinman1978}), or equivalently, recursive in the hyperjump; +and for this $\alpha$ the $\alpha$-recursive +(resp. $\alpha$-semi-recursive) subsets of $\omega$ are exactly the +the subsets recursive (resp. semi-recursive) in $\mathsf{E}_1$ +(\cite[chapter VIII, corollary 4.16 on p. 412]{Hinman1978}). \ordinal The smallest recursively hyperinaccessible ordinal: i.e., the smallest recursively inaccessible which is a limit of recursively inaccessibles. -\ordinal The smallest recursively Mahlo ordinal: i.e., the smallest -admissible ordinal $\alpha$ such that for any $\alpha$-recursive -function $f \colon \alpha \to \alpha$ there is an admissible -$\beta<\alpha$ which is closed under $f$. This is the smallest -ordinal $\alpha$ such that $L_\alpha \models \mathsf{KPM}$. +\ordinal\label{RecursivelyMahlo} The smallest recursively Mahlo +ordinal: i.e., the smallest admissible ordinal $\alpha$ such that for +any $\alpha$-recursive function $f \colon \alpha \to \alpha$ there is +an admissible $\beta<\alpha$ which is closed under $f$. This is the +smallest ordinal $\alpha$ such that $L_\alpha \models \mathsf{KPM}$. (Compare •\ref{CollapseMahlo}.) This is the smallest ordinal not the order type of a well-ordering @@ -199,16 +200,17 @@ partial normalization of the superjump, \cite[theorem 5 on Also note concerning this ordinal: \cite[corollary 9.4(ii) on p. 348]{RichterAczel1974}. -\ordinal The smallest $\Pi_3$-reflecting (``recursively weakly -compact'') ordinal. This can also be described as the smallest -``$2$-admissible'' ordinal: see \cite[theorem 1.16 on - p. 312]{RichterAczel1974}. (Compare •\ref{CollapseWeaklyCompact}.) +\ordinal\label{RecursivelyWeaklyCompact} The smallest +$\Pi_3$-reflecting (``recursively weakly compact'') ordinal. This can +also be described as the smallest ``$2$-admissible'' ordinal: +see \cite[theorem 1.16 on p. 312]{RichterAczel1974}. +(Compare •\ref{CollapseWeaklyCompact}.) -\ordinal The smallest $(+1)$-stable ordinal, i.e., the smallest -$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_{\alpha+1}$. This -is the smallest $\Pi^1_0$-reflecting (i.e., $\Pi_n$-reflecting for -every $n\in\omega$) ordinal: \cite[theorem 1.18 on p. 313 and - 333]{RichterAczel1974}. +\ordinal\label{WeaklyStable} The smallest $(+1)$-stable ordinal, i.e., +the smallest $\alpha$ such that $L_\alpha \mathrel{\preceq_1} +L_{\alpha+1}$. This is the smallest $\Pi^1_0$-reflecting (i.e., +$\Pi_n$-reflecting for every $n\in\omega$) ordinal: \cite[theorem 1.18 + on p. 313 and 333]{RichterAczel1974}. (Compare •\ref{CollapsePiTwoZeroIndescribable}.) @@ -245,6 +247,8 @@ the subsets recursive (resp. semi-recursive) in $\mathsf{E}_1^\#$ (combine \cite[theorem 1 on p. 313, theorem 2 on p. 318]{Aczel1970} and \cite[theorem D on p. 304]{RichterAczel1974}). +[\FINDTHIS: how stable is this ordinal?] + \ordinal The smallest $(^{++})$-stable ordinal, i.e., the smallest $\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_{\alpha^{++}}$ where $\alpha^+,\alpha^{++}$ are the two smallest admissible |