This commit is contained in:
parent
e59e97ca03
commit
f7f9d7d638
@ -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}$.
|
||||
|
||||
\begin{observe}
|
||||
\[
|
||||
\Tr \subseteq {2^{\N}}^{<\N}
|
||||
\]
|
||||
is closed (where we take the topology of the Cantor space).
|
||||
$\Tr \subseteq {2^{\N}}^{<\N}$ is closed
|
||||
(where we take the topology of the Cantor space).
|
||||
\end{observe}
|
||||
\gist{%
|
||||
Indeed, for any $ s \in \N^{<\N}$
|
||||
|
@ -33,7 +33,6 @@ with $(f^{-1}(\{1\}), <)$.
|
||||
and $[\alpha_i, \alpha_{i+1})$ to $(i,i+1)$.
|
||||
\end{proof}
|
||||
|
||||
% TODO ANKI-MARKER
|
||||
|
||||
\begin{definition}[\vocab{Kleene-Brouwer ordering}]
|
||||
Let $(A,<)$ be a linear order and $A$ countable.
|
||||
@ -58,25 +57,30 @@ with $(f^{-1}(\{1\}), <)$.
|
||||
$(T, <_{KB}\defon{T})$ is well ordered.
|
||||
\end{proposition}
|
||||
\begin{proof}
|
||||
If $T$ is ill-founded and $x \in [T]$,
|
||||
then for all $n$, we have $x\defon{n+1} <_{KB} x\defon{n}$.
|
||||
Thus $(T, <_{KB}\defon{T})$ is not well ordered.
|
||||
\gist{%
|
||||
If $T$ is ill-founded and $x \in [T]$,
|
||||
then for all $n$, we have $x\defon{n+1} <_{KB} x\defon{n}$.
|
||||
Thus $(T, <_{KB}\defon{T})$ is not well ordered.
|
||||
|
||||
Conversely, let $<\defon{KB}$ be not a well-ordering
|
||||
on $T$.
|
||||
Let $s_0 >_{KB} s_1 >_{KB} s_2 >_{KB} \ldots$
|
||||
be an infinite descending chain.
|
||||
We have that $s_0(0) \ge s_1(0) \ge s_2(0) \ge \ldots$
|
||||
stabilizes for $n > n_0$.
|
||||
Let $a_0 \coloneqq s_{n_0}(0)$.
|
||||
Now for $n \ge n_0$ we have that $s_n(0)$ is constant,
|
||||
hence for $n > n_0$ the value $s_{n}(1)$ must be defined.
|
||||
Thus there is $n_1 \ge n_0$ such that $s_n(1)$
|
||||
is constant for all $n \ge n_1$.
|
||||
Let $a_1 \coloneqq s_{n_1}(1)$
|
||||
and so on.
|
||||
Then $(a_0,a_1,a_2, \ldots) \in [T]$.
|
||||
Conversely, let $<\defon{KB}$ be not a well-ordering
|
||||
on $T$.
|
||||
Let $s_0 >_{KB} s_1 >_{KB} s_2 >_{KB} \ldots$
|
||||
be an infinite descending chain.
|
||||
We have that $s_0(0) \ge s_1(0) \ge s_2(0) \ge \ldots$
|
||||
stabilizes for $n > n_0$.
|
||||
Let $a_0 \coloneqq s_{n_0}(0)$.
|
||||
Now for $n \ge n_0$ we have that $s_n(0)$ is constant,
|
||||
hence for $n > n_0$ the value $s_{n}(1)$ must be defined.
|
||||
Thus there is $n_1 \ge n_0$ such that $s_n(1)$
|
||||
is constant for all $n \ge n_1$.
|
||||
Let $a_1 \coloneqq s_{n_1}(1)$
|
||||
and so on.
|
||||
Then $(a_0,a_1,a_2, \ldots) \in [T]$.
|
||||
}{easy}
|
||||
\end{proof}
|
||||
|
||||
% TODO ANKI-MARKER
|
||||
|
||||
\begin{theorem}[Lusin-Sierpinski]
|
||||
The set $\LO \setminus \WO$
|
||||
(resp.~$2^{\Q} \setminus \WO$)
|
||||
|
@ -4,6 +4,7 @@
|
||||
% TODO gist info
|
||||
% TODO link to long version (provide link to main document)
|
||||
|
||||
% TODO \phantomsection to cross link
|
||||
\newcommand{\gist}[2]{%
|
||||
\ifcsname EnableGist\endcsname%
|
||||
#2%
|
||||
|
Loading…
Reference in New Issue
Block a user