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\newpage
\subsubsection*{Introduction}

\begin{example*}
  $(\Zbb, +)$ is \emph{not} the same as $(\Zbb,
  +, \cdot)$.

  $(\Zbb, +)$ is \emph{decidable} (there exists an
  algorithm to decide whether a given sentence is
  true or false in the model).

  However, $(\Zbb, +, \cdot)$ is not decidable
  (G\"odel's completeness theorem).
\end{example*}

\begin{example*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item $(\Cbb, +, 0, 1)$
    \item $(V, +, f_r)$ (where $V$ is an
      $\Rbb$-vector space, and $f_r$ is the map $r
      : v \mapsto rv$ for any $r \in \Rbb$).
  \end{enumerate}
  These structures are both \emph{strongly
  minimal}.
\end{example*}

\begin{definition*}[Strongly minimal]
  \glsadjdefn{sm}{strongly minimal}{theory}%
  A theory is \emph{strongly minimal} if all
  formulas in one variable are either finite or
  co-finite.
\end{definition*}

For the $\Cbb$ example: formulas in
one variable are polynomial equations or
inequations, so solution set is always either
finite or cofinite (recall Fundamental Theorem of
Algebra).

For the vector space example: the formulas in one
variable are of the form $a \mathbf{x} =
\mathbf{b}$ or $a \mathbf{x} \neq
\mathbf{b}$.

Cheats:
\begin{itemize}
  \item Boolean combinations and quantifiers:
    \[
      \exists y axy^2 + y + x^2 + 3 = 0
    .\]
    Need \emph{quantifier elimination} (boolean
    combinations are easy to deal with).
  \item Elementary extensions (chapter 1).
\end{itemize}

Interestingly: strongly minimal structures all
carry notion of dimensions. For example:
\begin{itemize}
  \item In $(\Cbb, +, \cdot, 0, 1)$ this is
    transcendence degree.
  \item In $(V, +, f_r)_{r \in \Rbb}$, this is
    linear dimension.
\end{itemize}

If interested in further reading: see
\begin{center}
  \url{https://forkinganddividing.com/}
\end{center}

\setcounter{section}{-1}

\newpage
\section{Review of First Order Logic}

\subsection{Languages}

\[
  \ub{\mathcal{L}}_{\text{language}} =
  \ub{\mathcal{F}}_{\text{function symbols}} \cup
  \ub{\mathcal{R}}_{\text{relation symbols}} \cup
  \ub{\mathcal{C}}_{\text{constant symbols}}
.\]
\begin{example*}
  \phantom{}
  \begin{itemize}
    \item $\mathcal{L}_{\text{group}} = \{*, e\}$,
      with example sentences $x \cdot x = e$,
      $\exists y, x \cdot y = e$.
    \item $\mathcal{L}_{\text{rings}} = \{+,
      \times, 0, 1\}$: $x^2 + x + 1 = 0$.
    \item $\mathcal{L}_{\text{o}} = \{<\}$:
      $\forall x \forall y ((x < y \wedge y < x)
      \to x = y)$.
  \end{itemize}
\end{example*}

Convention: all languages include $=$.

\subsection{Structures}

\begin{definition*}
  Given a language $L$, an $L$-structure is a
  triple
  \[
    \mathcal{M} = \langle M, \hat{\mathcal{F}},
    \hat{\mathcal{R}}, \hat{\mathcal{C}} \rangle
  .\]
  $M$ is an underlying set.

  Convention: $M \neq \emptyset$.

  $\hat{\mathcal{F}}$: for every $n$-ary $f \in
  \mathcal{F}$ we have $\hat{f} \in
  \hat{\mathcal{F}}$ a function $\hat{f} : m^n \to
  m$.

  $\hat{\mathcal{R}}$: for every $n$-ary $R \in
  \mathcal{R}$, we have $\hat{R} \in
  \hat{\mathcal{R}}$, which is a subset of $M^n$.

  $\hat{\mathcal{C}}$: for every $c \in
  \mathcal{C}$, we have $\hat{c} \in
  \hat{\mathcal{C}}$, with $\hat{c} \in M$.
\end{definition*}

\begin{itemize}
  \item $\langle \Cbb, +, 0\rangle$ and $\langle
    \Cbb, \times, 1\rangle$ are both
    $\mathcal{L}_{\text{group}}$-structures.
  \item $\langle\Qbb, <\rangle$ and $\langle\Qbb,
    x + y = 3\rangle$ are both
    $\mathcal{L}_{\text{o}}$-structures.
\end{itemize}

\subsection{Formulas / sentences}

\begin{itemize}
  \item \emph{Terms}: made of variables, constant symbols
    and function symbols in a `sensible way'
    \[
      x + yx + 1 + 1
      \qquad \cancel{\cdot + x \mid \cdot \cdot}
    .\]
  \item \emph{Atomic formulas}: Plugging terms
    into one relation symbol
    \[
      x + yx + 1 + 1 = 0
      \qquad \cancel{== + 1 \cdot 0}
    \]
  \item \emph{Formulas}:
    \begin{itemize}
      \item Boolean combinations
        ($\neg, \wedge, \vee, \to, \leftrightarrow$)
      \item Quantifiers ($\exists, \forall$)
    \end{itemize}
    in a `sensible way':
    \[
      \exists y (x + yx + 1 + 1 = 0 \vee x = 1)
      \qquad \cancel{\forall x + \wedge 0}
    .\]

    A formula with $n$ free variables defines a
    subset of $M^n$.
\end{itemize}

\begin{example*}
  $\mathcal{L}_{\text{o}}$-structure
  $\langle \Qbb, <\rangle$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/ca9ff7cab94a46ee.png}
  \end{center}
\end{example*}

Formulas with no free variables are called
sentences.

In an $\mathcal{L}$-structure $M$, these are
either:
\begin{itemize}
  \item True: $\mathcal{M} \models \sigma$
  \item False: $\mathcal{M} \not\models \sigma$
\end{itemize}
In formula $\phi(\ol{x})$ with free variables, we
can plug a tuple $\ul{a} \in M^n$. We say $M$
satisfies $\phi(\ul{x})$ at $\ul{a}$, and we write
$\mathcal{M} \models \phi(\ul{a})$ (models /
satisfies) if
$\phi(\ul{a})$ is true in $\mathcal{M}$.

\begin{definition*}
  A set of sentences $\Sigma$ is satisfiable in
  $\mathcal{M}$ if
  for all $\sigma \in \Sigma$, $\mathcal{M}
  \models \sigma$.
\end{definition*}

\begin{theorem*}[Compactness Theorem]
  Let $\Sigma$ be a set of
  $\mathcal{L}$-sentences. $\Sigma$ is satisfiable
  if and only if every finite subset of $\Sigma$
  is satisfiable.
\end{theorem*}

($\Sigma$ is satisfiable if there is an
$\mathcal{L}$-structure $\mathcal{M}$ such that
$\Sigma$ is satisfiable in $\mathcal{M}$)

\begin{corollary*}[Upward L\"owenheim Skolem]
  Any theory that has either:
  \begin{itemize}
    \item arbitrary large finite models
    \item at least one infinite model
  \end{itemize}
  has arbitrarily large models.
\end{corollary*}
