Merge branch 'main' of https://git.abstractnonsen.se/josia-notes/w23-logic-3
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@ -20,7 +20,7 @@
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T \in \IF &\iff& \exists \beta \in \cN .~\forall n \in \N.~T(\beta\defon{n}) = 1.
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\end{IEEEeqnarray*}
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Consider $\{(T, \beta) \in \Tr \times \cN : \forall n.~ T(\beta\defon{n}) = 1\}$.
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Consider $D \coloneqq \{(T, \beta) \in \Tr \times \cN : \forall n.~ T(\beta\defon{n}) = 1\}$.
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Note that this set is closed in $\Tr \times \cN$,
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since it is a countable intersection of clopen sets.
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% TODO Why clopen?
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@ -123,9 +123,9 @@ For the proof we need some prerequisites:
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$\IF$ is $\Sigma^1_1$-complete.
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\end{corollary}
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\begin{proof}
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Let $A \subseteq X$ is analytic
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and $X$ Polish and uncountable,
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then
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Let $X$ be Polish.
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Suppose that $A \subseteq X$ is analytic and uncountable.
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Then
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% https://q.uiver.app/#q=WzAsNSxbMCwwLCJYIl0sWzEsMCwiXFxjTiJdLFsyLDAsIlxcVHIiXSxbMCwxLCJBIl0sWzEsMSwiYihBKSJdLFsxLDIsImYiXSxbMCwxLCJiIl0sWzMsMCwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbNCwxLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dXQ==
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\[\begin{tikzcd}
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X & \cN & \Tr \\
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@ -135,6 +135,7 @@ For the proof we need some prerequisites:
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\arrow[hook, from=2-1, to=1-1]
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\arrow[hook, from=2-2, to=1-2]
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\end{tikzcd}\]
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where $f$ is chosen as in \yaref{thm:lec12:1}.
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If $X$ is Polish and countable and $A \subseteq X$ analytic,
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just consider
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