diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex index e3d8b35..b631b42 100644 --- a/inputs/lecture_13.tex +++ b/inputs/lecture_13.tex @@ -39,7 +39,6 @@ is not a well-order on a countable set. Thus $\otp(\faktor{W}{\sim}, <) = \omega_1$. - \todo{move this} \end{remark} }{} diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex index 1b98bb8..dcfc009 100644 --- a/inputs/lecture_14.tex +++ b/inputs/lecture_14.tex @@ -280,9 +280,9 @@ We have shown (assuming \AxC to choose contained clubs): \end{refproof} \begin{remark}+ $\diagi_{\beta < \kappa} C_{\beta}$ actually - \emph{is} a club: - It suffices to show that $\diagi_{\beta < \kappa} C_\beta$ is closed. - This can be shown in the same way as for $\diagi_{\beta < \kappa} D_\beta$. + \emph{is} a club, + since $\diagi_{\beta < \kappa} C_\beta$ is closed, + again cf.~\yaref{rem:diagiclosed}. % Let $\lambda < \kappa$ be a limit ordinal. % Suppose that $\lambda \not\in \diagi_{\beta < \kappa} D_\beta$. % Then there exists $\alpha < \lambda$ such that diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index d919e49..d4ffb15 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -43,7 +43,6 @@ Recall that $F \subseteq \cP(\kappa)$ is a filter if $X,Y \in F \implies X \cap Y \in F$, $X \in F, X \subseteq Y \subseteq \kappa \implies Y \in F$ and $\emptyset \not\in F, \kappa \in F$. -\todo{Move this to the definition of filter?} }{} \begin{definition} A filter $F$ is an \vocab{ultrafilter} @@ -104,11 +103,7 @@ one cofinality. \begin{refproof}{thm:solovay}% \gist{% - %\footnote{``This is one of the arguments where it is certainly - % worth it to look at it again''} - % TODO: Look at this again and think about it. - % TODO TODO TODO - + \footnote{``This is one of the arguments where it is certainly worth it to look at it again.''} We will only prove this for $\aleph_1$. Fix $S \subseteq \aleph_1$ stationary. @@ -144,7 +139,7 @@ one cofinality. \end{claim} \begin{subproof} Otherwise for all $n < \omega$, - there is a $\delta$ such that + there is a $\delta$ such that $\{\alpha \in S^\ast : \gamma_n^{\alpha} > \delta\}$ is nonstationary. Let $\delta_n$ be the least such $\delta$. @@ -196,20 +191,23 @@ one cofinality. $\gamma_n^{\alpha} = \delta'$. Write $\delta_i = \delta'$ and $T_i = T$. - - \begin{claim} - \label{thm:solovay:p:c2} - Each $T_i$ is stationary - and if $i \neq j$, then $T_i \cap T_j = \emptyset$. - \footnote{maybe this should not be a claim} - \end{claim} - \begin{subproof} - The first part is true by construction. - Let $j < i$. - Then if $\alpha \in T_i$, $\alpha' \in T_j$, - we get $\gamma_n^{\alpha'} = \delta_j < \delta_i = \gamma_n^{\alpha}$ - hence $\alpha \neq \alpha'$. - \end{subproof} + + By construction, all the $T_i$ are stationary. + Since the $\delta_i$ are strictly increasing + and since $\gamma_n^{\alpha} = \delta_i$ for all $\alpha \in T_i$, + we have that the $T_i$ are disjoint. + % \begin{claim} + % \label{thm:solovay:p:c2} + % Each $T_i$ is stationary + % and if $i \neq j$, then $T_i \cap T_j = \emptyset$. + % \end{claim} + % \begin{subproof} + % The first part is true by construction. + % Let $j < i$. + % Then if $\alpha \in T_i$, $\alpha' \in T_j$, + % we get $\gamma_n^{\alpha'} = \delta_j < \delta_i = \gamma_n^{\alpha}$ + % hence $\alpha \neq \alpha'$. + % \end{subproof} Now let \[ @@ -232,8 +230,6 @@ one cofinality. \item $\exists n < \omega.~\forall \delta < \omega_1: \{\alpha \in S^\ast : \gamma^{\alpha}_n > \delta\}$ stationary: \begin{itemize} - % TODO THINK! - % TODO TODO TODO \item Otherwise $\forall n < \omega.~\exists \delta.~\{\alpha \in S^\ast : \gamma^{\alpha}_n > \delta\} $ nonstationary. \item $\delta_n\coloneqq $ least such $\delta$, $C_n$ club s.t.~$C_n \cap \{\alpha \in S^\ast : \gamma^{\alpha}_n > \delta_n\} = \emptyset$. diff --git a/inputs/lecture_17.tex b/inputs/lecture_17.tex index 0b0da9b..4ed2f66 100644 --- a/inputs/lecture_17.tex +++ b/inputs/lecture_17.tex @@ -1,9 +1,6 @@ \lecture{17}{2023-12-14}{Silver's Theorem} We now want to prove \yaref{thm:silver}. -% More generally, if $\kappa$ is a singular cardinal of uncountable cofinality -% such that $2^{\lambda} = \lambda^+$ for all $\lambda < \kappa$, -% then $2^{\kappa} = \kappa^+$. \gist{% \begin{remark} @@ -12,7 +9,7 @@ We now want to prove \yaref{thm:silver}. \end{remark} }{} -We will only proof +We will only prove \gist{% \yaref{thm:silver} in the special case that $\kappa = \aleph_{\omega_1}$ (see \yaref{thm:silver1}). @@ -153,6 +150,7 @@ We will only proof Let $\beta < \omega_1$ be such that for all $Y \in A_2$ and for all $\alpha \in T$, $h_Y(\alpha) = \beta$. + % TODO WHY DOES THIS WORK? Then $\overline{f}_Y(\alpha) < \aleph_\beta$ for all $Y \in A_2$ and $\alpha \in T$. diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex index 7499fbd..118ffc1 100644 --- a/inputs/lecture_18.tex +++ b/inputs/lecture_18.tex @@ -27,7 +27,6 @@ Since $2^{\lambda} < \kappa$, \AxPow works. The other axioms are trivial. - \todo{Exercise} \end{proof} \begin{corollary} $\ZFC$ does not prove the existence of inaccessible @@ -40,7 +39,7 @@ \end{proof} \begin{definition}[Ulam] - A cardinal $\kappa > \aleph_0$ is \vocab{measurable} + A cardinal $\kappa > \aleph_0$ is \vocab{measurable} iff there is an ultrafilter $U$ on $\kappa$, such that $U$ is not principal\gist{\footnote{% i.e.~$\{\xi\} \not\in U$ for all $\xi < \kappa$% diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 3795f80..0d58d26 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -79,6 +79,7 @@ and $\forall x \in y.~\phi$ abbreviates $\forall x.~x \in y \to \phi$. A similar arguments yields \vocab{upwards absoluteness} for $\Sigma_1$-formulas and \vocab{downwards absoluteness} for $\Pi_1$-formulas: \begin{lemma} + \label{lem:pi1downardsabsolute} Let $M$ be transitive. Let $\phi(x_0,\ldots,x_n) \in \cL_\in$ and $a_0,\ldots,a_n \in M$. Then diff --git a/inputs/lecture_22.tex b/inputs/lecture_22.tex index 5af0e30..04efc59 100644 --- a/inputs/lecture_22.tex +++ b/inputs/lecture_22.tex @@ -112,7 +112,7 @@ is well founded. The formula $\forall x.~\forall y .~((\forall z \in x.~z \in y \land \forall z \in y.~z \in x) \to x = y)$ is $\Pi_1$, hence it is true in $M[g]$ - by %TODO REF downward absolutenes. + by \yaref{lem:pi1downardsabsolute}. \item \AxFund: Again, \[ @@ -137,7 +137,7 @@ is well founded. so $M[g] \models \text{``$\{x,y\}$ is the pair of $x$ and $y$''}$. Hence $M[g] \models \AxPair$. \item \AxUnion: - Similar to \AxPair.\gist{\todo{Exercise}}{} + Similar to \AxPair. \end{itemize} \end{proof} @@ -194,7 +194,6 @@ Still missing are \[ p \Vdash^{\mathbb{P}}_M \phi(\tau_1,\ldots, \tau_k). \] - \end{enumerate} \end{theorem} \begin{proof}