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\begin{fcdefnstar}[Absolute formula]
  \glsadjdefn{absm}{absolute}{formula}%
  We say \emph{$\varphi$ is absolute for $M$} if
  for all $x_1, \ldots, x_n \in M$, we have
  \[
    M \models \varphi(x_1, \ldots, x_n)
    \iff \varphi(x_1, \ldots, x_n) \text{ is true}
  .\]
\end{fcdefnstar}

Clearly, if $\varphi$ is \gls{abs} for $M$ and
\gls{abs} for $N$, then it's \gls{abs} between $M$
and $N$.

Cohen's proof becomes:

If $M$ is a countable transitive set such that $M
\models \ZFC$, then there is a a countable
transitive set $N \supseteq M$ such that $N
\models \ZFC + \neg \CH$.

\newpage
\section{Transitive Models}

Observation: If $M$ is transitive and $M \models
\text{``$e$ is empty''}$, then $e = \emptyset$.
This is because if $w \in e$, then $w \in e$ and
$e \in M$ gives us that $w \in M$ by transitivity,
so $M \models w \in e$, so $M \models \text{``$e$
is not empty''}$.

\begin{fclemmastar}[]
  Assuming:
  - $M$ is transitive
  Then:
  \[
    M \models \text{Extensionality} +
    \text{Foundation}
  .\]
\end{fclemmastar}

\begin{proof}
  Extensionality: $\forall x \forall y (\forall w
  (w \in x \leftrightarrow w \in y) \to x = y)$. Take $x \neq
  y$, $x, y \in M$. Without loss of generality
  take $z \in x \setminus y$. Then $z \in x$ and
  $x \in M$ so $z \in M$. So $M \models z \in x
  \wedge z \notin y$. So $M \models \neg\forall w
  (w \in x \leftrightarrow w \in y)$.

  Foundation: $\forall x(x \neq \emptyset \to
  \exists m(m \in x \wedge \forall w \neg (w \in m
  \wedge w \in x)))$. Take $x \in M$. $M \models x
  \neq \emptyset$ so $x$ is not empty. So find $m
  \in x$ which is $\in$-minimal. Then since $x \in
  M$ as well, we have $m \in M$. Therefore $x$ has
  an $\in$-minimal element in $M$.
\end{proof}

\subsection{Absoluteness for transitive models}

\begin{fcdefnstar}[Bounded quantifier]
  We call a quantifier of the form $\exists x \in
  y, \varphi$ or $\forall x \in y, \varphi$ a
  \emph{bounded quantifier}.

  (Defined by $\exists x \in y, \varphi \defeq
  \exists x (x \in y \wedge \varphi)$ and $\forall
  x \in y \varphi \defeq \forall x (x \in y \to
  \varphi)$).
\end{fcdefnstar}

\begin{fcdefnstar}[Closed under bounded quantification]
  A class of formulas $\Gamma$ is \emph{closed
  under bounded quantification} if whenever
  $\varphi$ is in $\Gamma$, then so are $\exists x
  \in y, \varphi$ and $\forall x \in y, \varphi$.
\end{fcdefnstar}

\begin{fcdefnstar}[Delta0]
  \glssymboldefn{delz}%
  $\Delta_0$ is the smallest class of formulas
  containing the atomic formulas that is closed
  under propositional connectives and bounded
  quantifiers.

  Let $T$ be any theory. Then $\Delta_0^T$ is the
  class of formulas equivalent to a $\Delta_0$
  formula in $T$.
\end{fcdefnstar}

\begin{fcthmstar}[]
  $\delz$ formulas are \cloze{\gls{absm} for transitive
  models.}
\end{fcthmstar}

\begin{proof}
  By induction:
  \begin{enumerate}[(1)]
    \item All atomic formulas are \gls{absm} by the
      substructure lemma.
    \item Propositional connectives: exactly the
      same proof as in the substructure lemma.
    \item Assume that $\varphi$ is \gls{absm} and
      show that $\exists x \in y, \varphi$ and
      $\forall x \in y, \varphi$ are absolute.
      \begin{itemize}
        \item $\exists x \in y, \varphi$: If
          $\exists x \in y, \varphi$ is true for
          some $y \in M$, then pick a witness $x
          \in y$. Since $y \in M$, we have $x \in
          M$. By the induction hypothesis, we have
          that $M \models \varphi(x, y)$. Thus $M
          \models \exists x (x \in y \wedge
          \varphi(x, y))$.

          If $M \models \exists x(x \in y \wedge
          \varphi(x, y))$, then $x \in y \wedge
          \varphi(x, y)$ is true.
        \item $\forall x \in y, \varphi$: Similar.
          \qedhere
      \end{itemize}
  \end{enumerate}
\end{proof}

\begin{fccorostar}[]
  Assuming:
  - $T$ is any theory
  - $M \models T$ is transitive
  Then:
  $\delzt$-formulas are \gls{absm} for $M$.
\end{fccorostar}

\begin{fcdefnstar}[$Sigma1$, $Pi1$]
  \glssymboldefn{Pi0Sigma1}%
  A formula is called $\Sigma_1$ if it is of the
  form $\exists x_1, \ldots \exists x_n, \varphi$
  where $\varphi$ is $\delz$.

  It is called $\Pi_1$ if it is of the form
  $\forall x_1, \ldots, \forall x_n, \varphi$
  where $\varphi$ is $\delz$.
\end{fcdefnstar}

(same for $\Sigma_1^T$, $\Pi_1^T$).

% TODO
% \begin{fccorostar}[]
%   $\Sigma_1^T$-formulas are
%   \cloze[false]{\gls{uabs}} for this model and $\Pi_1^T$ are
%   \cloze{\gls{dabs}}.
% \end{fccorostar}

\begin{proof}
  Just definition of the semantics of $\exists,
  \forall$.
\end{proof}

\begin{example*}
  What is $\delz$?
  \begin{enumerate}
    \item $x \in y$
    \item $x = y$
    \item $x \subseteq y$: $\iff \forall w \in x
      (w \in y)$
    \item $z = \{x\}$: $\iff x \in z \wedge
      \forall w \in z (w = x)$
    \item $z = \{x, y\}$
    \item $z = (x, y) = \{\{x\}, \{x, y\}\}$
    \item $z = \emptyset$: $\iff \forall w \in z(w
      \neq w)$
    \item $z = x \cup y$: $\iff x \subseteq z
      \wedge y \subseteq z \wedge \forall w \in z
      (w \in x \vee w \in y)$
    \item $z = x \cap y$
    \item $z = x \setminus y$
    \item $z = x \cup \{x\}$
    \item $z \text{ is transitive}$
    \item $z = \bigcup x$
  \end{enumerate}
\end{example*}

\begin{fcdefnstar}[Absolute function]
  \glsadjdefn{absf}{absolute}{function}%
  Let $M$ a transitive set, and $F : M^n \to
  M$ (note that this means that $M$ is closed
  under $F$). We say $F$ is \emph{absolute for $M$} if
  there is a formula $\Phi$ which is \gls{absm}
  for $M$ such that
  \[
    F(x_1, \ldots, x_n) = z \iff \Phi(x_1, \ldots,
    x_n, z)
  .\]
\end{fcdefnstar}

Observation: So, if $M$ is closed under pairing
($\forall x, y \in M, \{x, y\} \in M$), then the
pairing operation $x, y \mapsto \{x, y\}$ is
\gls{absf}, and therefore $M \models
\text{Pairing}$.

Similarly for union.

\begin{fclemmastar}[]
  Assuming:
  - $\varphi$ is \gls{absm} for $M$
  - $F, G_1, \ldots, G_n$ are \gls{absf} operations
  on $M$
  Then:
  \begin{align*}
    \psi(x_1, \ldots, x_m)
    &\defeq \varphi(G_1(x_1, \ldots, x_m), \ldots,
    G_n(x_1, \ldots, x_m)) \\
    H(x_1, \ldots, x_m)
    &\defeq F(G_1(x_1, \ldots, x_m), \ldots,
    G_n(x_1, \ldots, x_m))
  \end{align*}
  are \gls{absm} for $M$.
\end{fclemmastar}

\begin{proof}
  Check the definitions!
\end{proof}

\begin{example*}
  More examples:
  \begin{enumerate}[(1)]
    \setcounter{enumi}{13}
    \item $z$ is an ordered pair:
      \[
        \exists s \in z, \exists d \in z, \exists x
        \in s, \exists y \in d, (\forall w \in s, (w
        = x) \wedge \forall v \in d, (v = x \wedge v
        = y) \wedge \forall w \in z, (w = s \vee w
        = d))
      .\]
    \item $z = a \times b$
    \item $z \text{ is a relation}$
    \item $z = \dom x$
    \item $z = \range x$
    \item $z \text{ is a function}$
    \item $z \text{ is injective}$
    \item $z \text{ is surjective}$
    \item $z \text{ is bijective}$
  \end{enumerate}
\end{example*}