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@ -173,10 +173,8 @@ i.e.~we want to associate a tree $T \subseteq \N^{<\N}$}%
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Let \vocab{$\Tr$} $ \coloneqq \{T \in {2^{\N}}^{<\N} : T \text{ is a tree}\} \subseteq {2^{\N}}^{<\N}$.
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Let \vocab{$\Tr$} $ \coloneqq \{T \in {2^{\N}}^{<\N} : T \text{ is a tree}\} \subseteq {2^{\N}}^{<\N}$.
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\begin{observe}
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\begin{observe}
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\[
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$\Tr \subseteq {2^{\N}}^{<\N}$ is closed
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\Tr \subseteq {2^{\N}}^{<\N}
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(where we take the topology of the Cantor space).
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\]
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is closed (where we take the topology of the Cantor space).
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\end{observe}
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\end{observe}
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\gist{%
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\gist{%
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Indeed, for any $ s \in \N^{<\N}$
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Indeed, for any $ s \in \N^{<\N}$
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@ -33,7 +33,6 @@ with $(f^{-1}(\{1\}), <)$.
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and $[\alpha_i, \alpha_{i+1})$ to $(i,i+1)$.
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and $[\alpha_i, \alpha_{i+1})$ to $(i,i+1)$.
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\end{proof}
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\end{proof}
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% TODO ANKI-MARKER
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\begin{definition}[\vocab{Kleene-Brouwer ordering}]
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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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Let $(A,<)$ be a linear order and $A$ countable.
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@ -58,6 +57,7 @@ with $(f^{-1}(\{1\}), <)$.
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$(T, <_{KB}\defon{T})$ is well ordered.
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$(T, <_{KB}\defon{T})$ is well ordered.
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\end{proposition}
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\end{proposition}
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\begin{proof}
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\begin{proof}
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\gist{%
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If $T$ is ill-founded and $x \in [T]$,
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If $T$ is ill-founded and $x \in [T]$,
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then for all $n$, we have $x\defon{n+1} <_{KB} x\defon{n}$.
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then for all $n$, we have $x\defon{n+1} <_{KB} x\defon{n}$.
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Thus $(T, <_{KB}\defon{T})$ is not well ordered.
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Thus $(T, <_{KB}\defon{T})$ is not well ordered.
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@ -76,7 +76,11 @@ with $(f^{-1}(\{1\}), <)$.
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Let $a_1 \coloneqq s_{n_1}(1)$
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Let $a_1 \coloneqq s_{n_1}(1)$
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and so on.
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and so on.
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Then $(a_0,a_1,a_2, \ldots) \in [T]$.
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Then $(a_0,a_1,a_2, \ldots) \in [T]$.
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}{easy}
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\end{proof}
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\end{proof}
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% TODO ANKI-MARKER
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\begin{theorem}[Lusin-Sierpinski]
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\begin{theorem}[Lusin-Sierpinski]
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The set $\LO \setminus \WO$
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The set $\LO \setminus \WO$
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(resp.~$2^{\Q} \setminus \WO$)
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(resp.~$2^{\Q} \setminus \WO$)
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@ -4,6 +4,7 @@
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% TODO gist info
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% TODO gist info
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% TODO link to long version (provide link to main document)
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% TODO link to long version (provide link to main document)
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% TODO \phantomsection to cross link
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\newcommand{\gist}[2]{%
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\newcommand{\gist}[2]{%
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\ifcsname EnableGist\endcsname%
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\ifcsname EnableGist\endcsname%
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#2%
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#2%
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