From 279b7057e982a279936f154d478aa44c5c39e642 Mon Sep 17 00:00:00 2001 From: zeramorphic Date: Fri, 9 Feb 2024 16:03:45 +0000 Subject: [PATCH] Minor fix Signed-off-by: zeramorphic --- iii/forcing/02_constructibility.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/iii/forcing/02_constructibility.tex b/iii/forcing/02_constructibility.tex index e8493d4..8949244 100644 --- a/iii/forcing/02_constructibility.tex +++ b/iii/forcing/02_constructibility.tex @@ -412,12 +412,12 @@ \subsection{Well-ordering the universe} \item \( <_{\alpha + 1}^0 \) is the well-ordering of \( \mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha} \) given by making \( \qty{\mathrm{L}_\alpha} \) the maximal element. \item Suppose that \( <_{\alpha + 1}^n \) is defined. We end-extend \( <_{\alpha + 1}^n \) to form \( <_{\alpha + 1}^{n + 1} \) as follows. - Suppose \( x, y \notin \mathcal D^n(\mathrm{L}_\alpha) \). + Suppose \( x, y \notin \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \). We say \( x <_{\alpha + 1}^{n+1} \) if either \begin{enumerate} - \item the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha) \) with \( x = \mathcal F_i(u, v) \) is less than the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha) \) with \( \mathcal y = \mathcal F_i(u, v) \); or - \item these indices \( i \) are equal, and the \( <_{\alpha + 1}^n \)-least \( u \in \mathcal D^n(\mathrm{L}_\alpha) \) such that there exists \( v \in \mathcal D^n(\mathrm{L}_\alpha) \) with \( x = \mathcal F_i(u, v) \) is less than the \( <_{\alpha + 1}^n \)-least \( u \in \mathcal D^n(\mathrm{L}_\alpha) \) such that there exists \( v \in \mathcal D^n(\mathrm{L}_\alpha) \) with \( y = \mathcal F_i(u, v) \); or - \item both of these coincide, and \( <_{\alpha + 1}^n \)-least \( v \in \mathcal D^n(\mathrm{L}_\alpha) \) with \( x = \mathcal F_i(u, v) \) is less than the least \( v \in \mathcal D^n(\mathrm{L}_\alpha) \) with \( y = \mathcal F_i(u, v) \). + \item the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( x = \mathcal F_i(u, v) \) is less than the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( \mathcal y = \mathcal F_i(u, v) \); or + \item these indices \( i \) are equal, and the \( <_{\alpha + 1}^n \)-least \( u \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) such that there exists \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( x = \mathcal F_i(u, v) \) is less than the \( <_{\alpha + 1}^n \)-least \( u \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) such that there exists \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( y = \mathcal F_i(u, v) \); or + \item both of these coincide, and \( <_{\alpha + 1}^n \)-least \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( x = \mathcal F_i(u, v) \) is less than the least \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( y = \mathcal F_i(u, v) \). \end{enumerate} \end{enumerate} \end{proof}