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Recap: $M$ a \gls{ctm} of $\zfc$ (or a
sufficiently large finite fragment).

$\Pbb \in M$: \gls{dense} / \gls{filt} / \dgen.

\begin{example*}
  $\Fn(\omega, 2)$ produces a new function $f :
  \omega \to 2$. Cohen forcing.
\end{example*}

\begin{example*}
  $\Fn(X, Y)$ produces a surjection $f : X \to Y$.
  $\Fn(\omega, Y)$ COLAPSE of $Y$.
\end{example*}

\begin{example*}
  $\Fn(X \times Y, 2)$ produces an injection $f :
  Y \to \mathcal{P}(X)$.
\end{example*}

If $M$ is a \gls{ctm} of $\zfc$, then
$\mathcal{D}_M \defeq \{D \subseteq P \text{
\gls{dense}} : D \in M\}$ is countable, so by
Theorem, we have a \dgen[\mathcal{D}_M].

\begin{fcdefn}[$P$-generic over $M$]
  We say $F$ is \emph{$\Pbb$-generic over $M$} if
  it is \dgen[\mathcal{D}_M]. These always exist
  if $M$ is a \gls{ctm}.
\end{fcdefn}

GOAL: Build an extension $M[G]$ such that $M
\subseteq M[G]$, $M[G]$ a \gls{ctm} of $\zfc$, $G
\in M[G]$ and $M[G]$ is minimal.

\subsubsection*{Names}

\textbf{Idea:} Think of elements of $\Pbb$ as
``truth values'' for the von Neumann construction.

\begin{align*}
  \Name_0^{\Pbb}
  &\defeq \emptyset \\
  \Name_{\alpha + 1}^{\Pbb}
  &\defeq \{\tau : \tau \subseteq
  \Name_\alpha \times \Pbb\} \\
  \Name_\lambda^{\Pbb}
  &\defeq \bigcup_{\alpha < \lambda}
  \Name_\alpha^{\Pbb}
\end{align*}
Then
\[
  \Name^{\Pbb}
  \defeq \bigcup_{\alpha \in \Ord}
  \Name_\alpha^{\Pbb}
\]
is the proper class of all names.

Consider: $\Pbb = \{0, 1\}$. Then $\Name^{\Pbb}
\cong V$.

Since this is a recursive definition using
\gls{abs} concepts, being a name is \gls{abs} for
transitive models:
\[
  \{\tau : M \models \text{$\tau$ is a $\Pbb$-name}\}
  = \Name^{\Pbb} \cap M
.\]

TODO

\subsubsection*{Examples}

$\emptyset \in \Name_1^{\Pbb}$,
\[
  \tau_p \defeq \{(\emptyset, p)\} \in
  \Name_2^{\Pbb}
\]
``The name for a set that contains $\emptyset$
with value $p$.''
\[
  \tau_{pq}
  \defeq \{(\tau_p, q)\}
.\]
``The name for whatever $\tau_p$ describes with
value $q$''.

\subsubsection*{Interpretation}

If $F \subseteq \Pbb$, we interpret a $\Pbb$-name
as follows:
\[
  \val(\tau, F)
  \defeq \{\val(\sigma, F) : \exists p \in F,
  (\sigma, p) \in \tau\}
.\]
\textbf{Important:} This is a recursive definition.

Thus: the valuation is absolute for transitive
models containing $\tau$ and $F$.

\begin{example*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item $\emptyset$: clearly $\val(\emptyset, F)
      = \emptyset$.
    \item $\tau_p$:
      \[
        \val(\tau_p, F)
        \defeq \begin{cases}
          \{\emptyset\} & p \in F \\
          \emptyset & p \notin F
        \end{cases}
      \]
    \item $\tau_{pq}$:
      \[
        \val(\tau_{pq}, F)
        \defeq \begin{cases}
          \emptyset & q \notin F \\
          \{\{\emptyset\}\} & q \in F \wedge p \in
          F \\
          \{\emptyset\} & q \in F \wedge p \notin
          F
        \end{cases}
      \]
      The relationship between $p$ and $q$ affects these
      possibilities.

      Example: if $q \le p$ and $F$
      is a \gls{filt}, then $\{\emptyset\}$ is
      impossible.

      Example: if $q = \indicator{}$
      and $F$ is a non-empty \gls{filt}, then
      $\emptyset$ is impossible.

      Example: if $p \perp q$ and $F$ is a
      \gls{filt}, then $\{\{\emptyset\}\}$ is
      impossible.
  \end{enumerate}
\end{example*}

\begin{fcdefn}[Generic extension]
  The \emph{(generic) extension} for any \gls{ctm}
  $M$ and any $F \subseteq \Pbb$ where $\Pbb \in
  M$ is
  \[
    M[F]
    \defeq \{\val(\tau, F) : \tau \in \Name^{\Pbb}
    \cap M\}
  .\]
\end{fcdefn}

Obviously, $M[F]$ is a countable set with
$\emptyset \in M[F]$ (Example (1)).

Also, by definition, $M[F]$ is transitive.

\textbf{Note:}
\[
  M[F] \models \text{Extensionality} +
  \text{Foundation}
.\]

\subsubsection*{Need to show}

\begin{enumerate}[(1)]
  \item $M \subseteq M[F]$.
  \item $F \in M[F]$.
  \item $M[F] \models \zfc$.
  \item $M[F]$ is minimal.
\end{enumerate}

\subsubsection*{Canonical Names}

\begin{fcdefn}[Canonical name]
  Let $x \in M$. Define by recursion the
  \emph{canonical name for $x$} by
  \[
    \check{w}
    \defeq \{(\check{y}, \indicator{}) : y \in x\}
  .\]
\end{fcdefn}

\begin{fclemma}[]
  $\val(\check{x}, F) = x$ if \cloze{$\indicator{} \in
  F$.}
\end{fclemma}

\begin{proof}
  Induction.
\end{proof}

\begin{fccoro}[]
  $M \subseteq M[F]$ if \cloze{$\indicator{} \in
  F$.}
\end{fccoro}

Alternative construction of canonical names
without $\indicator{}$ is on \es{2}.
\[
  \Gamma
  \defeq \{(\check{p}, p) : p \in \Pbb\}
.\]

\begin{fclemma}[]
  $\val(\Gamma, F) = F$.
\end{fclemma}

\begin{proof}
  Calculate:
  \begin{align*}
    \val(\Gamma, F)
    &= \{\val(\check{p}, F) : p \in F\} \\
    &= \{p : p \in F\}
    &&\text{(by previous lemma)}
    \\
    &= F
  \end{align*}
\end{proof}

\begin{corollary}[]
  $F \in M[F]$.
\end{corollary}

\begin{remark*}
  If $N$ is a \gls{ctm} with $M \subseteq N$ and
  $F \in N$, then $M[F] \subseteq N$. (by
  \gls{abs}ness of $\val(\tau, F)$).
\end{remark*}

Warm-up: Suppose $\sigma, \tau \in \Name^{\Pbb}$.
Define
\[
  \up(\sigma, \tau)
  \defeq \{(\sigma, \indicator{}), (\tau,
  \indicator{})\}
.\]

\textbf{Pairing:}
Then
\[
  \val(\up(\sigma, \tau), F))
  = \{\val(\sigma, F), \val(\tau, F)\}
.\]
(``up'' stands for unordered pair).

\begin{fccoro}[]
  $M[F] \models \text{Pairing}$ (if
  \cloze{$\indicator{} \in F$).}
\end{fccoro}

\textbf{Union:} If $\tau$ is a name, define
\[
  u_\tau
  \defeq \{(\sigma', r) : \exists \sigma, p, q,
  \text{s.t.}
  (\sigma, p) \in \tau, (\sigma', q) \in \sigma, r
  \le p, q\}
.\]
Claim: $\val(u_\tau, F) = \bigcup \val(\tau, F)$
if $F$ is a \gls{filt}.
\begin{proof}
  Suppose $z \in \val(u_\tau, F)$.

  So $z = \val(\sigma', F)$ for some $(\sigma', r)
  \in u_\tau$ with $r \in F$.

  So $\exists \sigma, p, q$ with $(\sigma, p) \in
  \tau$, $(\sigma', q) \in \sigma$, $r \le p, q$.

  So $p, q \in F$. Hence $\val(\sigma, F) \in
  \val(\tau, F)$, $z = \val(\sigma', F) \in
  \val(\sigma, F)$.

  So $z \in \bigcup \val(\tau, F)$.

  Conversely, suppose $z \in \bigcup \val(\tau,
  F)$. Then $\exists y, z \in y \in \val(\tau,
  F)$ ($z \to (\sigma', q) \in \sigma$ with $q \in
  F$, $y \to (\sigma, \rho) \in \tau$ with $p \in
  F$).

  Hence since $F$ a \gls{filt}, find $r \le p, q$
  with $r \in F$.

  Then $(\sigma', r) \in u_\tau$, so $z \in
  \val(u_\tau, F)$.
\end{proof}
