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\section{Recursive ordinals}

\setcounter{ordinalcnt}{-1}

\ordinal $0$ (zero).  This is the smallest ordinal, and the only one that is
neither successor nor limit.

\ordinal $1$ (one).  This is the smallest successor ordinal.

\ordinal $2$.

\ordinal $42$.

\ordinal $\omega$.  This is the smallest limit ordinal, and the
smallest infinite ordinal.

\ordinal $\omega+1$.  This is the smallest infinite successor ordinal.

\ordinal $\omega2$.

\ordinal $\omega^2$.

\ordinal $\omega^\omega$.

\ordinal $\omega^{\omega^\omega}$.

\ordinal $\varepsilon_0 = \varphi(1,0)$.  This is the limit of
$\omega, \omega^\omega, \omega^{\omega^\omega}, \ldots$, smallest
fixed point of $\xi \mapsto \omega^\xi$; in general, $\alpha \mapsto
\varepsilon_\alpha = \varphi(1,\alpha)$ is defined as the function
enumerating the fixed points of $\xi \mapsto \omega^\xi$.  This is the
proof-theoretic ordinal of Peano arithmetic.

\ordinal $\varepsilon_1 = \varphi(1,1)$.

\ordinal $\varepsilon_\omega$.

\ordinal $\varepsilon_{\varepsilon_0}$.

\ordinal $\varphi(2,0)$.  This is the limit of $\varepsilon_0,
\varepsilon_{\varepsilon_0}, \ldots$, smallest fixed point of $\xi
\mapsto \varepsilon_\xi$; in general, $\alpha \mapsto
\varphi(\gamma+1,\alpha)$ is defined as the function enumerating the
fixed points of $\xi \mapsto \varphi(\gamma,\xi)$.

\ordinal $\varphi(\omega,0)$.  This is the smallest ordinal closed
under primitive recursive ordinal functions.

\ordinal The Feferman-Schütte ordinal $\Gamma_0 = \varphi(1,0,0)$
(also $\psi(\Omega^{\Omega})$ for an appropriate collapsing
function $\psi$).  This is the limit of $\varepsilon_0, \penalty0
\varphi(\varepsilon_0,0), \penalty0 \varphi(\varphi(\varepsilon_0)),
\ldots$, smallest fixed point of $\xi \mapsto \varphi(\xi, 0)$.  This
is the proof-theoretic ordinal of $\mathsf{ATR}_0$.

\ordinal The Ackermann ordinal $\varphi(1,0,0,0)$ (also
$\psi(\Omega^{\Omega^2})$ for an appropriate collapsing
function $\psi$).

\ordinal The “small” Veblen ordinal ($\psi(\Omega^{\Omega^\omega})$ for
an appropriate collapsing function $\psi$).  This is the limit of
$\varphi(1,0), \penalty0 \varphi(1,0,0), \penalty0 \varphi(1,0,0,0),
\ldots$, the range of the Veblen functions with finitely many
variables.

\ordinal The “large” Veblen ordinal ($\psi(\Omega^{\Omega^\Omega})$
for an appropriate collapsing function $\psi$).  This is the range of
the Veblen functions with up to that many variables.

\ordinal The Bachmann-Howard ordinal ($\psi(\varepsilon_{\Omega+1}) =
\psi(\Omega_2)$ for an appropriate collapsing function $\psi$).  This
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\label{CollapseInaccessible} The collapse of an 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}.
(Compare •\ref{RecursivelyInaccessible}.)

\ordinal\label{CollapseMahlo} The collapse of a 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
(= $\Pi^1_1$-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
$\Pi^2_0$-indescribable cardinal.  This is the proof-theoretic ordinal
of $\mathsf{KP} + \Pi_\omega\hyphen\mathsf{Ref}$.
See \cite[part I]{Stegert2010}.  (Compare •\ref{WeaklyStable}.)

\ordinal The proof-theoretic ordinal of $\mathsf{Stability}$:
see \cite[part II]{Stegert2010}.

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\section{Recursively large countable ordinals}

\ordinal The Church-Kleene ordinal $\omega_1^{\mathrm{CK}}$: the
smallest admissible ordinal $>\omega$.  This is the smallest ordinal
which is not the order type of a recursive (equivalently:
hyperarithmetic) well-ordering on $\omega$.  The
$\omega_1^{\mathrm{CK}}$-recursive
(resp. $\omega_1^{\mathrm{CK}}$-semi-recursive) subsets of $\omega$
are exactly the $\Delta^1_1$ (=hyperarithmetic) (resp. $\Pi^1_1$)
subsets of $\omega$, and they are also exactly the subsets recursive
(resp. semi-recursive) in $\mathsf{E}$ (or $\mathsf{E}^\#$,
\CHECKTHIS).

\ordinal $\omega_\omega^{\mathrm{CK}}$: the smallest limit of
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\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
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\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
recursive in the superjump (\cite{AczelHinman1974} and
\cite{Harrington1974}); and for this $\alpha$ the $\alpha$-recursive
(resp. $\alpha$-semi-recursive) subsets of $\omega$ are exactly the
subsets recursive in the superjump (resp. semirecursive in the partial
normalization of the superjump, \cite[theorem 5 on
  p. 50]{Harrington1974}).

Also note concerning this ordinal: \cite[corollary 9.4(ii) on
  p. 348]{RichterAczel1974}.

\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\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}.)

\ordinal\label{PiOneOne} The smallest $(^+)$-stable ordinal, i.e., the
smallest $\alpha$ such that $L_\alpha \mathrel{\preceq_1}
L_{\alpha^+}$ where $\alpha^+$ is the smallest admissible
ordinal $>\alpha$.  This is the smallest $\Pi^1_1$-reflecting ordinal:
\cite[theorem 1.19 on p. 313 and 336]{RichterAczel1974}.  Also the sup
of the closure ordinals for $\Pi^1_1$ inductive operators:
\cite[theorem B on p. 303 or 10.7 on p. 355]{RichterAczel1974} and
\cite[theorem A on p. 222]{Cenzer1974}.

This is the smallest ordinal $\omega_1^{\mathsf{G}_1^\#}$ not the
order type of a well-ordering recursive in the nondeterministic
functional $\mathsf{G}_1^\#$ defined by $\mathsf{G}_1^\#(f) \approx
\{f(0)\}_{(\omega_1^f)^+}(f(1))$; and for this $\alpha$ the
$\alpha$-recursive (resp. $\alpha$-semi-recursive) subsets of $\omega$
are exactly the subsets recursive (resp. semi-recursive) in
$\mathsf{G}_1^\#$ (\cite[theorem 7.4 on p. 238]{Cenzer1974}).

\ordinal\label{SigmaOneOne} The smallest $\Sigma^1_1$-reflecting
ordinal.  Also the sup of the closure ordinals for $\Sigma^1_1$
inductive operators: \cite[theorem B on p. 303 or 10.7 on
  p. 355]{RichterAczel1974}.  That this ordinal is smaller than that
of •\ref{PiOneOne}: \cite[corollary 1 to theorem 6 on
  p.213]{Anderaa1974}; also see: \cite[theorem 6.5]{Simpson1978}.

This is the smallest ordinal $\omega_1^{\mathsf{E}_1^\#}$ not the
order type of a well-ordering recursive in the nondeterministic
version $\mathsf{E}_1^\#$ of the Tugué functional $\mathsf{E}_1$; and
for this $\alpha$ the $\alpha$-recursive
(resp. $\alpha$-semi-recursive) subsets of $\omega$ are exactly 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
ordinals $>\alpha$.  This is $\Sigma^1_1$-reflecting and greater than
the ordinal of •\ref{SigmaOneOne} \cite[theorem 6.4]{Simpson1978}
[\CHECKTHIS, probably similar to \cite[§6]{RichterAczel1974}].

\ordinal The smallest inaccessibly-stable ordinal, i.e., the smallest
$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_\beta$ where
$\beta$ is the smallest recursively inaccessible
(cf. •\ref{RecursivelyInaccessible}) ordinal $>\alpha$.

\ordinal The smallest Mahlo-stable ordinal, i.e., the smallest
$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_\beta$ where
$\beta$ is the smallest recursively Mahlo
(cf. •\ref{RecursivelyMahlo}) ordinal $>\alpha$.

\ordinal The smallest doubly $(+1)$-stable ordinal, i.e., the smallest
$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_\beta
\mathrel{\preceq_1} L_{\beta+1}$ (cf. •\ref{WeaklyStable}).

\ordinal\label{NonprojectibleStable} The smallest stable ordinal under
a nonprojectible ordinal, i.e., the smallest $\alpha$ such that
$L_\alpha \mathrel{\preceq_1} L_\beta$ where $\beta$ is the smallest
nonprojectible (the ordinal of •\ref{Nonprojectible}).

This is the smallest ordinal $\omega_1^{\mathbf{R}}$ not the order
type of a well-ordering recursive in a certain type $3$ functional
$\mathbf{R}$ defined in \cite{Harrington1975}; and for this $\alpha$
the $\alpha$-recursive subsets of $\omega$ are exactly the subsets
recursive in $\mathbf{R}$.  (See also \cite{John1986} and
\cite[example 4.10 on p. 369]{Simpson1978}.)

\ordinal\label{Nonprojectible} The smallest nonprojectible ordinal,
i.e., the smallest $\beta$ such that $\beta$ is a limit of
$\beta$-stable ordinals (ordinals $\alpha$ such that $L_\alpha
\mathrel{\preceq_1} L_\beta$ (cf. •\ref{NonprojectibleStable}); in
other words, the smallest $\beta$ such that $L_\beta \models
\mathsf{KPi}+$“the stable ordinals are unbounded”.  This is the
smallest ordinal $\beta$ such that $L_\beta \models
\mathsf{KP}\omega+\Sigma_1\hyphen\textsf{Sep}$ (cf. \cite[chapter V,
  theorem 6.3 on p. 175]{Barwise1975}), or such that $L_\beta \cap
\mathscr{P}(\omega)$ is a model of $\Pi^1_2$-comprehension
(cf. \cite[theorem VII.3.24 on p. 267 and theorem VII.5.17 on
  p. 292]{Simpson2009}).

In Jensen's terminology (\cite{Jensen1972}), this is the smallest
ordinal $\beta$ such that $\rho_1^\beta > \omega$, and in fact the
smallest $\beta>\omega$ such that $\rho_1^\beta = \beta$: that is, the
smallest ordinal $\beta$ such that every $\Sigma_1(L_\beta)$ subset
of $\omega$ is $\beta$-finite.  Sometimes also called the smallest
“strongly admissible” (or “strongly $\Sigma_1$-admissible”) ordinal.

\ordinal The smallest (weakly) $\Sigma_2$-admissible ordinal.  This is
the smallest ordinal $\beta$ such that $L_\beta \models
\mathsf{KP}\omega+\Delta_2\hyphen\textsf{Sep}$, or such that $L_\beta
\cap \mathscr{P}(\omega)$ is a model of $\Delta^1_3$-comprehension
(cf. \cite[theorem VII.3.24 on p. 267 and theorem VII.5.17 on
  p. 292]{Simpson2009}).

In Jensen's terminology (\cite{Jensen1972}), this is the smallest
ordinal $\beta$ such that $\eta_2^\beta > \omega$, and in fact the
smallest $\beta>\omega$ such that $\eta_2^\beta = \beta$: that is, the
smallest ordinal $\beta$ such that every $\Delta_2(L_\beta)$ subset
of $\omega$ is $\beta$-finite.

In the terminology of \cite[appendix]{MarekSrebrny1973}, this is the
first $\Delta_2$-gap ordinal.

\ordinal The ordinal of ramified analysis (often written $\beta_0$).
This is the smallest $\beta$ such that $L_\beta \models \bigwedge_n
\Sigma_n\hyphen\textsf{Sep}$ (the full separation scheme), or such
that $L_\beta \cap \mathscr{P}(\omega)$ is a model of full
second-order analysis (second-order comprehension), and in fact
$L_\beta \models \mathsf{ZFC}^-$ (that is, $\mathsf{ZFC}$ minus the
powerset axiom).

This starts the first gap in the constructible universe, and this gap
is of length $1$: see \cite{Putnam1963} and \cite[corollary 4.5 on
  p. 374]{MarekSrebrny1973}.

Note that this ordinal is $(+1)$-stable (cf. •\ref{WeaklyStable}) but
not $(+2)$-stable: \cite[corollary to theorem 6.14 on
  p. 384]{MarekSrebrny1973}.

\ordinal The start of the first gap of length $2$ in the constructible
universe.  If $\beta$ is this ordinal then $\beta$ is the $\beta$-th
gap ordinal (\cite[theorem 4.17 on p. 377]{MarekSrebrny1973}).

\ordinal The first ordinal $\beta$ which starts a gap of
length $\beta$ in the constructible universe.

\ordinal\label{OmegaOneSmallestModelKPWithOmegaOne} The ordinal $\beta
= \omega_1^{L_\alpha}$ where $\alpha$ is the smallest ordinal such
that $L_\alpha \models \mathsf{KP}+$“$\omega_1$ exists” (\CHECKTHIS:
$\alpha$ is the same as the first admissible $\alpha$ which is not
locally countable, because the existence of an uncountable set implies
the existence of an uncountable ordinal since Choice holds in
$L_\alpha$).  Then by construction $\beta$ starts a gap of length
$\alpha = \beta^+$ (the next admissible ordinal).

\ordinal The ordinal $\alpha$ mentioned
in •\ref{OmegaOneSmallestModelKPWithOmegaOne}.

[\FINDTHIS: is this the start of the first third-order gap in the
  constructible universe (\cite[§3]{MarekSrebrny1973})? that is, does
  the existence of $\omega_1$ give the existence of
  $\mathscr{P}(\omega)$ as $\{x \in L_{\omega_1} : x \subseteq
  \omega\}$?]

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\ordinal\label{OmegaOneSmallestModelZFC} The smallest uncountable
ordinal $\omega_1^{L_\alpha}$ in the smallest model $L_{\alpha}$
of $\mathsf{ZFC}$, assuming it exists (see •\ref{SmallestModelZFC}).
This ordinal is $\alpha$-stable.

\ordinal\label{SmallestModelZFC} The smallest ordinal $\alpha$ such
that $L_\alpha \models \mathsf{ZFC}$ (assuming it exists), i.e., the
height of the minimal model of $\mathsf{ZFC}$.

\ordinal\label{Stable} The smallest stable ordinal $\sigma$, i.e., the
smallest $\sigma$ such that $L_\sigma \mathrel{\preceq_1} L$, or
equivalently $L_\sigma \mathrel{\preceq_1} L_{\omega_1}$.  The set
$L_\sigma$ is the set of all $x$ which are $\Sigma_1$-definable in $L$
without parameters (\cite[chapter V, corollary 7.9(i) on
  p. 182]{Barwise1975}).

This ordinal is projectible to $\omega$ (i.e., in Jensen's
terminology), $\rho_1^\sigma = \omega$ (\cite[chapter V,
  theorem 7.10(i) on p. 183]{Barwise1975}).

This is the smallest ordinal $\delta^1_2$ which not the order type of
a well-ordering $\Delta^1_2$ on $\omega$; and in fact, for this
$\sigma = \delta^1_2$, the $\sigma$-recursive
(resp. $\sigma$-semi-recursive) subsets of $\omega$ are exactly the
$\Delta^1_2$ (resp. $\Sigma^1_2$) subsets of $\omega$
(\cite[chapter V, theorem 8.2 on p. 189 and corollary 8.3 on
  p. 191]{Barwise1975}).

This is also the smallest $\Sigma^1_2$-reflecting ordinal
(\cite{Richter1975}).

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\bibitem[RichterAczel1974]{RichterAczel1974} Wayne Richter \& Peter
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\bibitem[Stegert2010]{Stegert2010} Jan-Carl Stegert, \textit{Ordinal
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\end{thebibliography}

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