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Josia Pietsch 2024-01-24 23:13:56 +01:00
parent e59e97ca03
commit f7f9d7d638
Signed by: josia
GPG key ID: E70B571D66986A2D
3 changed files with 25 additions and 22 deletions

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@ -173,10 +173,8 @@ i.e.~we want to associate a tree $T \subseteq \N^{<\N}$}%
Let \vocab{$\Tr$} $ \coloneqq \{T \in {2^{\N}}^{<\N} : T \text{ is a tree}\} \subseteq {2^{\N}}^{<\N}$. Let \vocab{$\Tr$} $ \coloneqq \{T \in {2^{\N}}^{<\N} : T \text{ is a tree}\} \subseteq {2^{\N}}^{<\N}$.
\begin{observe} \begin{observe}
\[ $\Tr \subseteq {2^{\N}}^{<\N}$ is closed
\Tr \subseteq {2^{\N}}^{<\N} (where we take the topology of the Cantor space).
\]
is closed (where we take the topology of the Cantor space).
\end{observe} \end{observe}
\gist{% \gist{%
Indeed, for any $ s \in \N^{<\N}$ Indeed, for any $ s \in \N^{<\N}$

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@ -33,7 +33,6 @@ with $(f^{-1}(\{1\}), <)$.
and $[\alpha_i, \alpha_{i+1})$ to $(i,i+1)$. and $[\alpha_i, \alpha_{i+1})$ to $(i,i+1)$.
\end{proof} \end{proof}
% TODO ANKI-MARKER
\begin{definition}[\vocab{Kleene-Brouwer ordering}] \begin{definition}[\vocab{Kleene-Brouwer ordering}]
Let $(A,<)$ be a linear order and $A$ countable. Let $(A,<)$ be a linear order and $A$ countable.
@ -58,6 +57,7 @@ with $(f^{-1}(\{1\}), <)$.
$(T, <_{KB}\defon{T})$ is well ordered. $(T, <_{KB}\defon{T})$ is well ordered.
\end{proposition} \end{proposition}
\begin{proof} \begin{proof}
\gist{%
If $T$ is ill-founded and $x \in [T]$, If $T$ is ill-founded and $x \in [T]$,
then for all $n$, we have $x\defon{n+1} <_{KB} x\defon{n}$. then for all $n$, we have $x\defon{n+1} <_{KB} x\defon{n}$.
Thus $(T, <_{KB}\defon{T})$ is not well ordered. Thus $(T, <_{KB}\defon{T})$ is not well ordered.
@ -76,7 +76,11 @@ with $(f^{-1}(\{1\}), <)$.
Let $a_1 \coloneqq s_{n_1}(1)$ Let $a_1 \coloneqq s_{n_1}(1)$
and so on. and so on.
Then $(a_0,a_1,a_2, \ldots) \in [T]$. Then $(a_0,a_1,a_2, \ldots) \in [T]$.
}{easy}
\end{proof} \end{proof}
% TODO ANKI-MARKER
\begin{theorem}[Lusin-Sierpinski] \begin{theorem}[Lusin-Sierpinski]
The set $\LO \setminus \WO$ The set $\LO \setminus \WO$
(resp.~$2^{\Q} \setminus \WO$) (resp.~$2^{\Q} \setminus \WO$)

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@ -4,6 +4,7 @@
% TODO gist info % TODO gist info
% TODO link to long version (provide link to main document) % TODO link to long version (provide link to main document)
% TODO \phantomsection to cross link
\newcommand{\gist}[2]{% \newcommand{\gist}[2]{%
\ifcsname EnableGist\endcsname% \ifcsname EnableGist\endcsname%
#2% #2%