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4 changed files with 14 additions and 11 deletions
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@ -93,7 +93,7 @@ $\ZFC$ consists of the following axioms:
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The axiom of infinity says that there exists and inductive set.
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\end{axiom}
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\begin{axiomscheme}[Separation]
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\begin{axiomschema}[\vocab{Separation}]
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% TODO :(Aus)
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Let $\phi$ be some fixed
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fist order formula in $\cL_\in$.
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@ -110,7 +110,7 @@ $\ZFC$ consists of the following axioms:
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\[
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\forall a.~\exists b.~(b = \{x \in a; \phi(x)\}).
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\]
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\end{axiomscheme}
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\end{axiomschema}
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\begin{notation}
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\todo{$\cap, \setminus, \bigcap$}
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@ -126,21 +126,22 @@ $\ZFC$ consists of the following axioms:
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\item $\forall a.~\exists b.~(b = \bigcap a)$.
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\end{itemize}
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\end{remark}
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\begin{axiomscheme}[\vocab{Replacement} (Fraenkel)]
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\begin{axiomschema}[\vocab{Replacement} (Fraenkel)]
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Let $\phi$ be some $\cL_{\in }$ formula.
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Then
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\[
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\forall v_1 .~\exists b.~\forall y.~(y \in b \iff \exists x .~(x \in a \land \phi(x,y,v_1,v_p))).
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\]
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\end{axiomscheme}
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\end{axiomschema}
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\begin{axiom}[\vocab{Choice}]
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Every family of non-empty sets has a \vocab{choice set}:
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\todo{TODO}
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\[
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\forall x.~(\forall y \in x.~x \neq \emptyset \land
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\forall y \in x \forall y' \in x .(y \neq y' \implies x \cap y' = \emptyset) \implies \exists z .~\forall y \in x .~\exists u.~(z \cap y = \{u\})).
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\]
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\begin{IEEEeqnarray*}{rCl}
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\forall x .~&(&\\
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&& ((\forall y \in x.~y \neq \emptyset) \land (\forall y \in x .~\forall y' \in x .~(y \neq y' \implies y \cap y' = \emptyset))\\
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&& \implies\exists z.~\forall y \in x.~\exists u.~(z \cap y = \{u\})\\
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&)&
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\end{IEEEeqnarray*}
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\end{axiom}
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@ -283,7 +283,7 @@
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\end{claim}
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\begin{subproof}
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Suppose that $\dom(r) \subsetneq a$
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and $\ran(r) \susbsetneq b$.
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and $\ran(r) \subsetneq b$.
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Let $x \coloneqq \min(a \setminus \dom(r))$
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and $y \coloneqq \min(b\setminus \ran(r))$.
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@ -93,6 +93,8 @@
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\hypersetup{colorlinks, citecolor=violet, urlcolor=blue!80!black, linkcolor=red!50!black, pdfauthor=\@author, pdftitle=\ifdef{\@course}{\@course}{\@title}}
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\NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning}
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\NewFancyTheorem[thmtools = { style = thmredmarginandfill} , group = { big } ]{axiom}
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\NewFancyTheorem[thmtools = { style = thmredmarginandfill, name = Axiom Schema} , group = { big } ]{axiomschema}
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\DeclareSimpleMathOperator{ran} % TODO: ran vs range
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\DeclareSimpleMathOperator{range} % TODO
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@ -27,7 +27,7 @@
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\input{inputs/lecture_01}
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\input{inputs/lecture_02}
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\input{inputs/lecture_03}
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% \input{inputs/lecture_04}
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\input{inputs/lecture_04}
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\input{inputs/lecture_05}
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