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diff --git a/ordinal-zoo.tex b/ordinal-zoo.tex index 48659ed..e3a506d 100644 --- a/ordinal-zoo.tex +++ b/ordinal-zoo.tex @@ -430,6 +430,20 @@ $\Delta^1_2$ (resp. $\Sigma^1_2$) subsets of $\omega$ This is also the smallest $\Sigma^1_2$-reflecting ordinal (\cite{Richter1975}). +\bigbreak + +\textbf{\textcolor{orange}{Note:}} This document should probably not +start listing large cardinals, because \textbf{(0)} the fact that one +implies the other nonwithstanding, this is about “ordinals”, not +“cardinals”, \textbf{(1)} they are already well covered elsewhere +(see, e.g., \cite{Kanamori1997}) and \textbf{(2)} we don't want to +start making assumptions, e.g., about whether $\omega_1^L$ is or is +not equal to $\omega_1$, but without making such assumptions it is no +longer possible to correctly order definitions. Perhaps a median way +would be to assume $V=L$ for ordering, forget about measurable +cardinals and whatnot, and still include inaccessibles, Mahlo, weakly +compact, etc. + % % % @@ -587,6 +601,10 @@ so $A \in L_\gamma$ with $\gamma$ countable, as asserted. $R$-operator and the ordinal $\sigma_3$”, \textit{J. Symbolic Logic} \textbf{51} (1986), 1–11. +\bibitem[Kanamori1997]{Kanamori1997} Akihiro Kanamori, \textit{The + Higher Infinite} (corrected first edition), Perspectives in + Mathematical Logic, Springer-Verlag (1997), ISBN 3-540-57071-3. + \bibitem[MarekSrebrny1973]{MarekSrebrny1973} Wiktor Marek \& Marian Srebrny, “Gaps in the Constructible Universe”, \textit{Ann. Math. Logic} \textbf{6} (1974), 359–394. |