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Fix typo
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Signed-off-by: zeramorphic <[email protected]>
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zeramorphic committed Jan 29, 2024
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Expand Up @@ -579,7 +579,7 @@ \subsection{The reflection theorem}
Suppose that \( \bigwedge_{i=1}^n \varphi_i \) proves every axiom of \( T \).
By reflection, \( T \) proves that for every \( \alpha \) there is \( \beta > \alpha \) such that the \( \varphi_i \) hold in \( \mathrm{V}_\beta \) if and only if they hold in \( \mathrm{V} \).
Since they hold in \( \mathrm{V} \), they must hold in some \( \mathrm{V}_\beta \).
Fix \( \beta_0 \) to be the least ordinal such that \( \bigwedge_{i=1}^n \varphi_i^_{\mathrm{V}_{\beta_0}} \).
Fix \( \beta_0 \) to be the least ordinal such that \( \bigwedge_{i=1}^n \varphi_i^{\mathrm{V}_{\beta_0}} \).
Then all of the axioms of \( T \) hold in \( \mathrm{V}_{\beta_0} \), so \( \mathrm{V}_{\beta_0} \vDash T \).
Since \( T \) extends \( \mathsf{ZF} \), our basic absoluteness results hold, so in particular, if \( \alpha \in \mathrm{V}_{\beta_0} \) then
\[ \mathrm{V}_\alpha^{\mathrm{V}_{\beta_0}} = \mathrm{V}_\alpha \cap \mathrm{V}_{\beta_0} = \mathrm{V}_\alpha \]
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