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Fix typo
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zeramorphic committed Feb 9, 2024
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Expand Up @@ -559,18 +559,18 @@ \subsection{Combinatorial properties}
\end{remark}
\begin{definition}
Let \( \kappa \) be an uncountable cardinal.
Then the \emph{square principle} \( \square_\kappa \) is the assertion that there exists a sequence \( (C_\alpha) \) indexed by the limit ordinals \( \alpha \) in \( \kappa^+ \), such that
Then the \emph{square principle} \( \mdwhtsquare_\kappa \) is the assertion that there exists a sequence \( (C_\alpha) \) indexed by the limit ordinals \( \alpha \) in \( \kappa^+ \), such that
\begin{enumerate}
\item \( C_\alpha \) is a club subset of \( \alpha \);
\item if \( \beta \) is a limit ordinal of \( C_\alpha \) then \( C_\beta = C_\alpha \cap \beta \); then
\item if \( \cf(\alpha) < \kappa \) then \( \abs{C_\alpha} < \kappa \).
\end{enumerate}
\end{definition}
\begin{theorem}[Jensen]
If \( \mathrm{V} = \mathrm{L} \), then \( \square_\kappa \) holds for every uncountable cardinal \( \kappa \).
If \( \mathrm{V} = \mathrm{L} \), then \( \mdwhtsquare_\kappa \) holds for every uncountable cardinal \( \kappa \).
\end{theorem}
\begin{lemma}
If \( \square_{\omega_1} \), then there exists a stationary set \( S \subseteq \qty{\beta \in \omega_2 \mid \cf(\beta) = \omega} \) such that for all \( \alpha \in \omega_2 \) with \( \cf(\alpha) = \omega_1 \), \( S \cap \alpha \) is not stationary in \( \alpha \).
If \( \mdwhtsquare_{\omega_1} \), then there exists a stationary set \( S \subseteq \qty{\beta \in \omega_2 \mid \cf(\beta) = \omega} \) such that for all \( \alpha \in \omega_2 \) with \( \cf(\alpha) = \omega_1 \), \( S \cap \alpha \) is not stationary in \( \alpha \).
\end{lemma}
\begin{remark}
If \( \kappa \) is a weakly compact cardinal, then every stationary subset of \( \kappa \) \emph{reflects}: there is \( \alpha \in \kappa \) such that \( S \cap \alpha \) is stationary in \( \alpha \).
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