diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex index 542af7a..7ff994c 100644 --- a/inputs/lecture_10.tex +++ b/inputs/lecture_10.tex @@ -13,8 +13,24 @@ Applications of induction and recursion: Once we have such a function, $\{x\} \cup \bigcup \ran(F)$ is a set as desired. - - \todo{insert formal application of recursion theorem} + To get this $F$ using the recursion theorem, + pick $D$ such that + \[ + (\emptyset, 0, \{x\}) \in D + \] + and + \[ + (f, n+1, \bigcup\bigcup \ran(f)) \in D. + \] + The recursion theorem then gives a function + such that + \begin{IEEEeqnarray*}{rCl} + F(0) &=& \{x\},\\ + F(n+1) &=& \bigcup\bigcup \ran(F\defon{n+1})\\ + &=& \bigcup \bigcup \{\{x\}, x, \bigcup x, \ldots, \underbrace{\bigcup^{n-1} x}_{F(n)}\} + = \bigcup F(n), + \end{IEEEeqnarray*} + i.e.~$F(n+1) = \bigcup F(n)$. \end{proof} \begin{notation} @@ -26,7 +42,27 @@ Applications of induction and recursion: such that $F(\alpha) = \bigcup \{\cP(F(\beta)): \beta < \alpha\}$. \end{lemma} \begin{proof} - \todo{TODO} + Use the recursion theorem with $R = \in $ + and $(w,x,y) \in D$ iff + \[ + y = \bigcup \{\cP(\overline{y}) : \overline{y} \in \ran(w)\}. + \] + This function has the following properties: + \begin{IEEEeqnarray*}{rCl} + F(0) &=& \bigcup \emptyset = \emptyset,\\ + F(1) &=& \bigcup \{\cP(\emptyset)\} = \bigcup \{\{\emptyset\} \} = \{\emptyset\},\\ + F(2) &=& \bigcup \{\cP(\emptyset), \cP(\{\emptyset\})\} = \bigcup \{\{\emptyset\}, \{\emptyset, \{\emptyset\} \} \} = \{\emptyset, \{\emptyset\} \},\\ + \ldots + \end{IEEEeqnarray*} + + It is easy to prove by induction: + \begin{enumerate}[(a)] + \item Every $F(\alpha)$ is transitive. + \item $F(\alpha) \subseteq F(\beta)$ for all $\alpha \le \beta$. + \item $F(\alpha+1) = \cP(F(\alpha))$ for all $\alpha \in \OR$. + \item $F(\lambda) = \bigcup \{F(\beta) :\beta < \lambda\}$ + for $\lambda \in \OR$ a limit. + \end{enumerate} \end{proof} \begin{notation} Usually, one write $V_\alpha$ for $F(\alpha)$. @@ -37,4 +73,97 @@ Applications of induction and recursion: such that $x \in V_\alpha$, i.e.~$V = \bigcup \{V_{\alpha} : \alpha \in \OR\}$. \end{lemma} +\begin{proof} + We use induction on the well-founded $\in$-relation. + Let $A = \bigcup \{V_\alpha : \alpha \in \OR\}$. + We need to show that $A = V$. + By induction it suffices to prove that for every $x \in V$, + if $\{y : y \in x\} \subseteq A$, then $x \in A$. + The hypothesis says that for all $y \in x$, + there is some $\alpha$ with $y \in V_\alpha$. + Write $\alpha_y$ for the least such $\alpha$. + By \AxRep, $\{\alpha_y : y \in x\}$ + is a set and we may let + $\alpha = \sup \{\alpha_y : y \in x\} \ge \alpha_y$ + for all $y \in x$. + Then $y \in V_{\alpha_y} \subseteq V_\alpha$ + for all $y \in x$. + + In other words $x \subseteq V_\alpha$, + hence $x \in V_{\alpha+1}$. +\end{proof} + +\begin{lemma}[\vocab{Transitive collapse}/\vocab{Mostowski collapse}] + \yalabel{Mostowski Collapse}{Mostowski}{lem:mostowski} + Let $R$ be a binary set-like relation on a class $A$. + Then $R$ is well-founded iff there is a transitive class $B$ + such that + \[ + (B, \in\defon{B}) \cong (A, R), + \] + i.e.~there is an isomorphism $F$, + that is a function $F\colon B \to A$ + with $x \in y \iff (F(x),F(y)) \in R$ for $x,y \in B$. +\end{lemma} +\begin{proof} + ``$\impliedby$'' Suppose that $R$ is ill-founded + (i.e.~not well-founded). + Then there is some $(y_n : n < \omega)$ such that $y_n \in A$ + and $(y_{n+1}, y_n) \in R$ for all $n < \omega$. + But then if $F$ is an isomorphism as above, + \[ + F^{-1}(Y_{n+1}) \in F^{-1}(Y_n) + \] + for all $n < \omega$ $\lightning$ + + ``$\implies$ '' Suppose that $R$ is well-founded. + We want a transitive class $B$ and a function $F\colon B \leftrightarrow A$ + such that + \[ + x \in y \iff (F(x), F(y)) \in R. + \] + Equivalently $G\colon A \leftrightarrow B$ + with $(x,y) \in R$ iff $G(x) \in G(y)$ for all $x,y \in A$. + + In other words, $G(y) = \{G(x) : (x,y) \in R\}$. + Such a function $G$ and class $B$ exist by the recursion theorem. +\end{proof} + +\begin{lemma}[\vocab{Rank function}] + Let $R$ be a well-founded and set-like binary relation + on a class $A$. + Then there is a function $F\colon A \to \OR$, + such that for all $x,y \in A$ + \[(x,y) \in R \implies F(x) < F(y) F(x) < F(y).\] +\end{lemma} +\begin{proof} + By the recursion theorem, + there is $F$ such that + \[ + F(y) = \sup \{F(x) + 1 : (x,y) \in R\}. + \] + This function is as desired. +\end{proof} + +This does not skip any ordinals, +as $F(y)$ is the least ordinal $> F(x)$ +for all $(x,y) \in R$. +Thus $\ran(F)$ is transitive. +So either $\ran(F) = \OR$ +or $\ran(F) \in \OR$. +This $F$ is called the \vocab{rank function} for $(A, R)$. +\begin{notation} + \[ + \rk_R(x) = \|x\|_R \coloneqq F(x), + \] + and + \[ + \rk_R \rank(R) \coloneqq \ran(F). + \] + + In the special case that $R$ is a linear order on $A$, + hence a well-order, + $\rank(R)$ is called the \vocab{order type} of $R$ + (or of $(A,R)$), written $\otp(R)$. +\end{notation} diff --git a/inputs/lecture_11.tex b/inputs/lecture_11.tex index d8b597b..95f5879 100644 --- a/inputs/lecture_11.tex +++ b/inputs/lecture_11.tex @@ -158,21 +158,21 @@ so $|a| = \aleph_\beta$ for some $\beta \le |\alpha|$. \[ \ran(\Gamma\defon{\aleph_\alpha \times \aleph_\alpha}) \supseteq \aleph_\alpha, \] - as otherwise $\Gamma\defon{\aleph_\alpha \times \alepha_\alpha}: \alepha_{\alpha} \times \alepha_\alpha \to \eta$ - would be a bijection for some $\eta < \alepha_\alpha$, - but $\alepha_\alpha$ is a cardinal. + as otherwise $\Gamma\defon{\aleph_\alpha \times \aleph_\alpha}: \aleph_{\alpha} \times \aleph_\alpha \to \eta$ + would be a bijection for some $\eta < \aleph_\alpha$, + but $\aleph_\alpha$ is a cardinal. - Suppose that $\ran(\Gamma\defon{\aleph_\alpha \times \alepha_\alpha}) \supsetneq \aleph_\alpha$. + Suppose that $\ran(\Gamma\defon{\aleph_\alpha \times \aleph_\alpha}) \supsetneq \aleph_\alpha$. Then there exist $\eta, \eta' < \aleph_\alpha$ with \[ - \Gamma((\eta, \eta')) = \alepha_\alpha. + \Gamma((\eta, \eta')) = \aleph_\alpha. \] So $\Gamma\defon{\{(\gamma,\delta) : (\gamma,\delta) <^\ast (\eta, \eta'\}}$ - is bijective onto $\alepha_\alpha$. + is bijective onto $\aleph_\alpha$. If $(\gamma,\delta) <^\ast (\eta, \eta')$, then $\max \{\gamma,\delta\} \le \max \{\eta, \eta'\}$. - Say $\eta \le \eta' < \alepha_\alpha$ + Say $\eta \le \eta' < \aleph_\alpha$ and let $\aleph_\alpha = |\eta'|$. There is a surjection \[f\colon \underbrace{(\eta +1)}_{ \le \aleph_\beta} \times \underbrace{(\eta' + 1)}_{\sim \aleph_\beta} \twoheadrightarrow \aleph_\alpha.\] diff --git a/logic.sty b/logic.sty index 441909d..af1feb3 100644 --- a/logic.sty +++ b/logic.sty @@ -135,5 +135,9 @@ \newcommand{\concat}{{}^\frown} \DeclareMathOperator{\hght}{height} +\DeclareSimpleMathOperator{rank} +\DeclareSimpleMathOperator{rk} +\DeclareSimpleMathOperator{otp} + \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}}