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some fix for some problem
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Josia Pietsch 2024-01-25 20:25:21 +01:00
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inputs/lecture_14.tex
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@ -17,13 +17,13 @@
\] \]
and similarly for $<^\ast$. and similarly for $<^\ast$.
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJZIl0sWzIsMCwiWCJdLFswLDEsIkMnIl0sWzIsMSwiXFxiZWdpbnthcnJheX17Y31DXFxcXFxcUGlfMV4xLVxcdGV4dHtjb21wbGV0ZX1cXGVuZHthcnJheX0iXSxbMCwxLCJmIl0sWzIsMCwiXFxzdWJzZXRlcSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzMsMSwiXFxzdWJzZXRlcSIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV1d % https://q.uiver.app/#q=WzAsNCxbMCwwLCJZIl0sWzIsMCwiWCJdLFswLDEsIkMnIl0sWzIsMSwiXFxiZWdpbnthcnJheX17Y31DXFxcXFxcUGlfMV4xLVxcdGV4dHtjb21wbGV0ZX1cXGVuZHthcnJheX0iXSxbMCwxLCJmIl0sWzIsMCwiXFxzdWJzZXRlcSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzMsMSwiXFxzdWJzZXRlcSIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV1d
\begin{tikzcd} % \begin{tikzcd}
Y && X \\ % Y && X \\
{C'} && {\begin{array}{c}C\\\Pi_1^1-\text{complete}\end{array}} % {C'} && {\begin{array}{c}C\\\Pi_1^1-\text{complete}\end{array}}
\arrow["f", from=1-1, to=1-3] % \arrow["f", from=1-1, to=1-3]
\arrow["\subseteq", hook, from=2-1, to=1-1] % \arrow["\subseteq", hook, from=2-1, to=1-1]
\arrow["\subseteq"', hook, from=2-3, to=1-3] % \arrow["\subseteq"', hook, from=2-3, to=1-3]
\end{tikzcd} % \end{tikzcd}
Let $X = 2^{\Q} \supseteq \WO$. Let $X = 2^{\Q} \supseteq \WO$.
We have already show that $\WO$ is $\Pi^1_1$-complete. We have already show that $\WO$ is $\Pi^1_1$-complete.