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@ -59,12 +59,11 @@
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Take $A \in \Sigma^1_1(X) \setminus \cB(X)$%
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\footnote{e.g.~\yaref{thm:universals11}}
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and $f\colon X \to Y$ Borel.
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But then we get that $f^{-1}(B)$ is Borel $\lightnig$.
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But then we get that $f^{-1}(B)$ is Borel $\lightning$.
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}{.}
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\end{itemize}
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\end{observe}
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% TODO ANKI-MARKER
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\begin{theorem}
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\label{thm:lec12:1}
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Suppose that $A \subseteq \cN$ is analytic.
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@ -1,11 +1,12 @@
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\lecture{13}{2023-11-08}{}
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\gist{%
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% Recap
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$\LO = \{x \in 2^{\N\times \N} : x \text{ is a linear order}\} $.
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$\LO \subseteq 2^{\N \times \N}$ is closed
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and $\WO = \{x \in \LO: x \text{ is a wellordering}\} $
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is coanalytic in $\LO$.
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% End Recap
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}{}
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Another way to code linear orders:
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@ -32,6 +33,8 @@ with $(f^{-1}(\{1\}), <)$.
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and $[\alpha_i, \alpha_{i+1})$ to $(i,i+1)$.
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\end{proof}
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% TODO ANKI-MARKER
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\begin{definition}[\vocab{Kleene-Brouwer ordering}]
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Let $(A,<)$ be a linear order and $A$ countable.
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We define the linear order $<_{KB}$ on $A^{<\N}$
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