From 69bdc01f6c8274db0349be0e454296ff4c8d2542 Mon Sep 17 00:00:00 2001 From: zeramorphic Date: Mon, 29 Jan 2024 13:51:08 +0000 Subject: [PATCH] Fix typo Signed-off-by: zeramorphic --- iii/forcing/01.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/iii/forcing/01.tex b/iii/forcing/01.tex index 19457ab..2bc0d68 100644 --- a/iii/forcing/01.tex +++ b/iii/forcing/01.tex @@ -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 \[ \mathrm{V}_\alpha^{\mathrm{V}_{\beta_0}} = \mathrm{V}_\alpha \cap \mathrm{V}_{\beta_0} = \mathrm{V}_\alpha \]