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lecture 04
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Josia Pietsch 2023-11-05 00:50:26 +01:00
parent 340e034fdf
commit 430ad82e8b
Signed by: josia
GPG key ID: E70B571D66986A2D
4 changed files with 14 additions and 11 deletions
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19
inputs/lecture_04.tex
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@ -93,7 +93,7 @@ $\ZFC$ consists of the following axioms:
The axiom of infinity says that there exists and inductive set. The axiom of infinity says that there exists and inductive set.
\end{axiom} \end{axiom}
\begin{axiomscheme}[Separation] \begin{axiomschema}[\vocab{Separation}]
% TODO :(Aus) % TODO :(Aus)
Let $\phi$ be some fixed Let $\phi$ be some fixed
fist order formula in $\cL_\in$. fist order formula in $\cL_\in$.
@ -110,7 +110,7 @@ $\ZFC$ consists of the following axioms:
\[ \[
\forall a.~\exists b.~(b = \{x \in a; \phi(x)\}). \forall a.~\exists b.~(b = \{x \in a; \phi(x)\}).
\] \]
\end{axiomscheme} \end{axiomschema}
\begin{notation} \begin{notation}
\todo{$\cap, \setminus, \bigcap$} \todo{$\cap, \setminus, \bigcap$}
@ -126,21 +126,22 @@ $\ZFC$ consists of the following axioms:
\item $\forall a.~\exists b.~(b = \bigcap a)$. \item $\forall a.~\exists b.~(b = \bigcap a)$.
\end{itemize} \end{itemize}
\end{remark} \end{remark}
\begin{axiomscheme}[\vocab{Replacement} (Fraenkel)] \begin{axiomschema}[\vocab{Replacement} (Fraenkel)]
Let $\phi$ be some $\cL_{\in }$ formula. Let $\phi$ be some $\cL_{\in }$ formula.
Then Then
\[ \[
\forall v_1 .~\exists b.~\forall y.~(y \in b \iff \exists x .~(x \in a \land \phi(x,y,v_1,v_p))). \forall v_1 .~\exists b.~\forall y.~(y \in b \iff \exists x .~(x \in a \land \phi(x,y,v_1,v_p))).
\] \]
\end{axiomscheme} \end{axiomschema}
\begin{axiom}[\vocab{Choice}] \begin{axiom}[\vocab{Choice}]
Every family of non-empty sets has a \vocab{choice set}: Every family of non-empty sets has a \vocab{choice set}:
\todo{TODO} \begin{IEEEeqnarray*}{rCl}
\[ \forall x .~&(&\\
\forall x.~(\forall y \in x.~x \neq \emptyset \land && ((\forall y \in x.~y \neq \emptyset) \land (\forall y \in x .~\forall y' \in x .~(y \neq y' \implies y \cap y' = \emptyset))\\
\forall y \in x \forall y' \in x .(y \neq y' \implies x \cap y' = \emptyset) \implies \exists z .~\forall y \in x .~\exists u.~(z \cap y = \{u\})). && \implies\exists z.~\forall y \in x.~\exists u.~(z \cap y = \{u\})\\
\] &)&
\end{IEEEeqnarray*}
\end{axiom} \end{axiom}

2
inputs/lecture_05.tex
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@ -283,7 +283,7 @@
\end{claim} \end{claim}
\begin{subproof} \begin{subproof}
Suppose that $\dom(r) \subsetneq a$ Suppose that $\dom(r) \subsetneq a$
and $\ran(r) \susbsetneq b$. and $\ran(r) \subsetneq b$.
Let $x \coloneqq \min(a \setminus \dom(r))$ Let $x \coloneqq \min(a \setminus \dom(r))$
and $y \coloneqq \min(b\setminus \ran(r))$. and $y \coloneqq \min(b\setminus \ran(r))$.

2
logic.sty
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@ -93,6 +93,8 @@
\hypersetup{colorlinks, citecolor=violet, urlcolor=blue!80!black, linkcolor=red!50!black, pdfauthor=\@author, pdftitle=\ifdef{\@course}{\@course}{\@title}} \hypersetup{colorlinks, citecolor=violet, urlcolor=blue!80!black, linkcolor=red!50!black, pdfauthor=\@author, pdftitle=\ifdef{\@course}{\@course}{\@title}}
\NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning} \NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning}
\NewFancyTheorem[thmtools = { style = thmredmarginandfill} , group = { big } ]{axiom}
\NewFancyTheorem[thmtools = { style = thmredmarginandfill, name = Axiom Schema} , group = { big } ]{axiomschema}
\DeclareSimpleMathOperator{ran} % TODO: ran vs range \DeclareSimpleMathOperator{ran} % TODO: ran vs range
\DeclareSimpleMathOperator{range} % TODO \DeclareSimpleMathOperator{range} % TODO

2
logic2.tex
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@ -27,7 +27,7 @@
\input{inputs/lecture_01} \input{inputs/lecture_01}
\input{inputs/lecture_02} \input{inputs/lecture_02}
\input{inputs/lecture_03} \input{inputs/lecture_03}
% \input{inputs/lecture_04} \input{inputs/lecture_04}
\input{inputs/lecture_05} \input{inputs/lecture_05}