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authorDavid A. Madore <david+git@madore.org>2013-01-31 18:08:04 (GMT)
committerDavid A. Madore <david+git@madore.org>2013-01-31 18:08:04 (GMT)
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Gaps in the constructible universe.
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@@ -273,7 +273,7 @@ $\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_\beta
\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 first
+$L_\alpha \mathrel{\preceq_1} L_\beta$ where $\beta$ is the smallest
nonprojectible (the ordinal of •\ref{Nonprojectible}).
This is the smallest ordinal $\omega_1^{\mathbf{R}}$ not the order
@@ -290,8 +290,8 @@ $\beta$-stable ordinals (ordinals $\alpha$ such that $L_\alpha
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 $L_\beta \models
-\mathsf{KP}+\Sigma_1\hyphen\textsf{Sep}$ (cf. \cite[chapter V,
- theorem 6.3 on p. 175]{Barwise1975}), or such that $L_\alpha \cap
+\mathsf{KP}\omega+\Sigma_1\hyphen\textsf{Sep}$ (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}).
@@ -299,13 +299,80 @@ smallest ordinal $\beta$ such that $L_\beta \models
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 $\Sigma_1$ subset of $\omega$
-is $\beta$-finite.
+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 $\{x \in L_{\omega_1} : x \subseteq
+ \omega\}$?]
%
%
%
+\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
@@ -319,10 +386,11 @@ terminology), $\rho_1^\sigma = \omega$ (\cite[chapter V,
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
-$\sigma$ 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}).
+$\sigma = \delta^1_2$, 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}).
@@ -388,6 +456,14 @@ This is also the smallest $\Sigma^1_2$-reflecting ordinal
$R$-operator and the ordinal $\sigma_3$”, \textit{J. Symbolic Logic}
\textbf{51} (1986), 1–11.
+\bibitem[MarekSrebrny1973]{MarekSrebrny1973} Wiktor Marek \& Marian
+ Srebrny, “Gaps in the Constructible Universe”,
+ \textit{Ann. Math. Logic} \textbf{6} (1974), 359–394.
+
+\bibitem[Putnam1963]{Putnam1963} Hilary Putnam, “A Note on
+ Constructible Sets of Integers”, \textit{Notre Dame J. Formal Logic}
+ \textbf{4} (1963), 270–273.
+
\bibitem[Rathjen1990]{Rathjen1990} Michael Rathjen, “Ordinal Notations
Based on a Weakly Mahlo Cardinal”, \textit{Arch. Math. Logic}
\textbf{29} (1990), 249–263.
@@ -416,8 +492,8 @@ This is also the smallest $\Sigma^1_2$-reflecting ordinal
355–390.
\bibitem[Simpson2009]{Simpson2009} Stephen G. Simpson,
- \textit{Subsystems of Second-Order Arithmetic}, Perspectives in
- Logic, ASL (2009), ISBN 978-0-521-88439-6.
+ \textit{Subsystems of Second-Order Arithmetic} (second edition),
+ Perspectives in Logic, ASL (2009), ISBN 978-0-521-88439-6.
\bibitem[Stegert2010]{Stegert2010} Jan-Carl Stegert, \textit{Ordinal
Proof Theory of Kripke-Platek Set Theory Augmented by Strong