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\section*{model theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_theory}{}\paragraph*{{Model theory}}\label{model_theory}

[[!include model theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Scope}{Scope}\dotfill \pageref*{Scope} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{FirstOrderLogic}{Classical Model theory in First-order logic}\dotfill \pageref*{FirstOrderLogic} \linebreak
\noindent\hyperlink{definition+theory+and+model+fol}{Definition and remarks}\dotfill \pageref*{definition+theory+and+model+fol} \linebreak
\noindent\hyperlink{OperadsMonadsAndLawvereTheories}{Operads, Monads and Lawvere theories}\dotfill \pageref*{OperadsMonadsAndLawvereTheories} \linebreak
\noindent\hyperlink{HigherOrderLogic}{Higher-order logic}\dotfill \pageref*{HigherOrderLogic} \linebreak
\noindent\hyperlink{CategoricalModelTheory}{Infinitary logic and Categorical model theory}\dotfill \pageref*{CategoricalModelTheory} \linebreak
\noindent\hyperlink{important_theorems}{Important theorems}\dotfill \pageref*{important_theorems} \linebreak
\noindent\hyperlink{goedel_completeness_theorem}{Goedel completeness theorem}\dotfill \pageref*{goedel_completeness_theorem} \linebreak
\noindent\hyperlink{goedels_compactness_theorem}{Goedel's compactness theorem}\dotfill \pageref*{goedels_compactness_theorem} \linebreak
\noindent\hyperlink{o_ultraproduct_theorem}{o ultraproduct theorem}\dotfill \pageref*{o_ultraproduct_theorem} \linebreak
\noindent\hyperlink{corollaries_worth_thinking_about}{Corollaries worth thinking about}\dotfill \pageref*{corollaries_worth_thinking_about} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

On the one hand, there is [[internal language|syntax]]. On the other hand, there is [[structure|semantics]]. Model theory is (roughly) about the relations between the two: model theory studies classes of [[models]] of [[theories]], hence classes of ``[[structures in model theory|mathematical structures]]''.

A model theory for a particular [[logic]] typically works within a given [[universe]], and specifies a notion of ``[[mathematical structure]]'' in this context, namely a \emph{[[structure in model theory|structure for a language]]}, and a \emph{[[definition]] of [[truth]]}. Logic is typically specified by [[signature (in logic)|language]] (function symbols, relations symbols and constants), and formulas (formed from symbols, [[variables]], Boolean operations and [[quantifiers]]). Language together with a choice of a set of formulas without [[free variables]] (viewed as [[axioms]]) is called a [[theory]]. A structure is an interpretation of a language $L$ via a given set $M$ together with interpretation of the symbols of the language. A \textbf{[[model]]} of a [[theory]] $\mathcal{T}$ in the language $L$ is an [[structure in model theory|L-structure]] which satisfies each formula in $\mathcal{T}$.

The two main problems of model theory are

\begin{itemize}%
\item classification of a given theory $\mathcal{T}$ in a language $L$
\item study the family of all [[definable sets]] of a [[structure (in model theory)|structure]] $M$ of a [[language (in model theory)|language]] $L$

\end{itemize}
In all memorable examples, the collection of structures for a language will form an interesting [[category]] (see \hyperlink{BekeRosicky11}{Beke-Rosicky 11} for a characterization of these), and the subcollection of those structures verifying a given collection $Th$ of [[propositions]] in the language are an interesting [[subcategory]] again.

Model theory as currently conceived has strong analogies with the classical theory of [[algebraic varieties]] and hence with the part of [[algebraic geometry]] that deals with these. One such analogy is that [[definable subsets]] are likened to [[zero loci]] of [[equations]], and another consists of various notions of [[dimension]] which can be likened to Krull dimension. One can also say that classical algebraic geometry often provides a testing ground for more general developments in model theory. (For the most part, model theory does not deal however with more global concepts of modern algebraic geometry such as [[sheaves]] or [[schemes]].)

\hypertarget{Scope}{}\subsection*{{Scope}}\label{Scope}

In full generality, \emph{model theory} would study all kinds of [[models]] over all kinds of [[theories]], hence pretty much everything that is considered (in) [[mathematics]]. However, in order to find effective classification results for models, one traditionally restricts attention to very special kinds of theories only.

Traditionally the default subject of model theory (e.g. (\hyperlink{Hodges93}{Hodges 93})) are [[first-order theories]] (only) and their models in [[Set]] (only), these are the traditional [[structures in model theory]]. For emphasis the study of this specific case is also called \emph{first-order model theory} or \emph{classical model theory} (\hyperlink{SEPEntry}{SEP}). Among the fundamental classfication theorems provable in this context are the [[compactness theorem]] and the [[Löwenheim-Skolem theorem]]. (See also \emph{[[geometric stability theory]]}.)

More modern developments in model theory consider also wider classes of [[theories]] than just [[first-order theories]] (e.g \hyperlink{TentZiegler12}{Tent-Ziegler 12}). In particular there is \hyperlink{CategoricalModelTheory}{categorical model theory} (\hyperlink{MakkaiPare89}{Makkai-Par\'e{} 89}, \hyperlink{BekeRosicky11}{Beke-Rosicky 11}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Classical model theory is concerned with models of [[theories]] in [[first-order logic]], this is what we discuss first, below in

\begin{itemize}%
\item \hyperlink{FirstOrderLogic}{First-order logic}.

\end{itemize}
More generally one can consider models over theories in [[higher-order logic]], this we discuss below in

\begin{itemize}%
\item \hyperlink{HigherOrderLogic}{Higher-order logic}.

\end{itemize}
\hypertarget{FirstOrderLogic}{}\subsubsection*{{Classical Model theory in First-order logic}}\label{FirstOrderLogic}

See also/first [[theory]]/[[first-order theory]].

\textbf{Caveat Lector.} This may duplicate/contradict other nLab accounts of [[FOL]], e.g. [[theory]]; it is present here for illustrative purposes only. We attempt to obviate the trouble of quantifier scope by using \emph{addressing} rather than \emph{naming} of variables; specifically, the variable $x_n$ occurs \emph{bound} in a formula if it is nested within more than $n$ quantifiers, and otherwise \emph{free}.

We describe an [[operad]] $L$ in [[Set]] over two [[type]]s, $W$ and $P$ for ``words'' and ``propositions'', respectively. The full operad is freely generated by various disjoint suboperads

\begin{itemize}%
\item a free suboperad $O$ of operations, or functions, of types $W^n\to W$;
\item an $\mathbb{N}$-indexed suboperad $X$ of variables of types $1\to W$;
\item a (free) suboperad $R$ of predicates, or relations, of types $W^n\to P$;
\item the equality operad, of type $W^2\to P$
\item the boolean operad, $B$: it's got $\wedge, \vee,\neg,\dots$ of types $P^{k}\to P$ for $k=1,2$, and perhaps $k=0$ if you want to include $\bot$; these compose the way you think they should.
\item the quantifier $\forall:$, usually written $\forall x_j$, where $x_j\in X_0$; $\backslash$forall is of type $P\to P$.

\end{itemize}
Notable subsets of $L$ include $O[X]$, generated by $O\cup X$, the suboperad of parametrized words, and $R[X]$, the elements of types $? \to P$ in the suboperad generated by $O,R,X$.

Abusively, an \emph{$L$-structure}, or \emph{interpretation} of the functions and relations is an [[algebra over an operad|algebra]] $(W,P,O,R)\mapsto (M,2,O_M,R_M)$ for the suboperad $Q$ of $L$ generated by $O\cup R$ with the type $P$ interpreted by the initial [[boolean algebra]] $2$.

The \emph{Tarski Definition of Truth} is a [[functor|natural]] extension of the $Q$-algebra $M$ to an $L$-algebra, $(M,\dots)$ such that * for $\Phi$ in $R[X]$ and for all functions $m:X\to M$, the sensible thing $\Phi[m]$ is true in $M$. * $\Phi \wedge \Theta$ is true if for all functions $m:X\to M$, both $\Phi[m]$ and $\Theta[m]$ are true, and $\neg \Phi$ is true if and only if $\Phi$ is \emph{not}. * $(\forall:\Phi)[m]$ is true in $M$ precisely when $\Phi[m_a]$ is true for all $a\in M$, where

\begin{displaymath}
m_a(x_i) = \left\{ \itexarray{ a & i=0 \\ m(x_{i-1}) & else }\right.
\end{displaymath}
(Again, this really should be written more clearly, but it's a start.)

[[JCMcKeown]]: is there some nicer way to say the quantifier nonsense? I'm thinking along the lines

\begin{quote}%
there are two actions of $\mathbb{N}$ on $L$: one shifts all the variables by 1, the other adds a $\forall$ quantifier; and the Tarski extension is the one that makes these commute somehow


\end{quote}
\emph{ibid}: maybe it's more right to say that $P_M$ should be the boolean algebra $\mathcal{P}(|M|^\mathbb{N})$? This again has that natural action of $\mathbb{N}$ on it\ldots{}

\hypertarget{definition+theory+and+model+fol}{}\paragraph*{{Definition and remarks}}\label{definition+theory+and+model+fol}

An $L$-structure $M$, as an $L$-algebra with extra properties, defines a \textbf{complete first-order theory} $Th(M)$, that subset of $L$ which $M$ interprets as $\top\in 2$, or \emph{true}. Conversely, given a collection $T$ of elements of $L$ of type $?\to P$, we say that $M\models T$, or in words $M$ \textbf{is a model of the theory} $T$ whenever $T\subset Th(M)$. There is an obvious Galois connection between theories $T$ and the collections of $L$-structures that are models. Much of deeper model theory studies the fine structure of this connection.

\hypertarget{OperadsMonadsAndLawvereTheories}{}\subsubsection*{{Operads, Monads and Lawvere theories}}\label{OperadsMonadsAndLawvereTheories}

Structures from [[universal algebra]], [[higher algebra]] and [[categorical logic]] that conveniently allow tospeak about [[theories]] and their [[models]] are [[operads]], [[monads]] and [[Lawvere theories]]. Each of these may be understood as characterizing a [[theory]]. Its [[models]] then are the corresponging algebras, see at \emph{[[algebra over an operad]]}, \emph{[[algebra over a monad]]} and \emph{[[algebra over a Lawvere theory]]}, respectively.

\hypertarget{HigherOrderLogic}{}\subsubsection*{{Higher-order logic}}\label{HigherOrderLogic}

Remaining within [[Set]], we can also generalize beyond first-order logic to various [[higher-order logic|higher-order logics]].

((insert your favourite variant here))

\hypertarget{CategoricalModelTheory}{}\subsubsection*{{Infinitary logic and Categorical model theory}}\label{CategoricalModelTheory}

For a reasonable fragment of [[infinitary logic]], then the [[category]] of all [[models]] over [[theories]] in this [[language]] is an [[accessible category]] and moreover all accessible categories arise this way. (\hyperlink{MakkaiPare89}{Makkai-Pare 89}, \hyperlink{BekeRosicky11}{Beke-Rosicky 11}).

\hypertarget{important_theorems}{}\subsection*{{Important theorems}}\label{important_theorems}

The following are closely interrelated, and depend on having a suitable universe $V$. We can view them as theorems of $ZFC$ or as (relatively mild) conditions on $V$.

\emph{(\ldots{} clarify \ldots{})}

\hypertarget{goedel_completeness_theorem}{}\paragraph*{{Goedel completeness theorem}}\label{goedel_completeness_theorem}

Given a first-order theory $T$ in some language $L$, $T$ is consistent iff there is a model of $T$ in $V$ --- that is, iff $M\models T$ for some $M\in V$.

See at \emph{[[completeness theorem]]}.

\hypertarget{goedels_compactness_theorem}{}\paragraph*{{Goedel's compactness theorem}}\label{goedels_compactness_theorem}

Under the same hypotheses, $T$ is consistent iff every finite subset of $T$ is consistent; expressed semantically, a theory $T$ has a model iff every finite subset of $T$ has a model.

See at \emph{[[compactness theorem]]}.

\hypertarget{o_ultraproduct_theorem}{}\paragraph*{{o ultraproduct theorem}}\label{o_ultraproduct_theorem}

\emph{(\ldots{}think of a good way to state this\ldots{})}

\hypertarget{corollaries_worth_thinking_about}{}\paragraph*{{Corollaries worth thinking about}}\label{corollaries_worth_thinking_about}

It follows that first-order theories are quite permissive; or in other words that they're inefficient at pinning down particular structures.

For example, consider the complete first-order theory $Th(V_\omega,\in)$, and any total order $(X,\lt)$. If one \emph{expands} the language (coresponding to an injective morphism of operads) to include constant symbols $c_x$ for $x\in X$, then for any subset $s$ of $X$ of finite size $n + 1$, one has

\begin{displaymath}
(V_\omega,\in,0,1,\cdots,n)\models Th(V_\omega,\in)\cup \{c_x\in c_y \mid x\lt y;x,y\in s\}
\end{displaymath}
so that the \emph{finite extensions} of $Th(V_\omega,\in)$ by suborders of $X$ are all consistent; by compactness, the fully extended theory $Th(V_\omega,\in)\cup \{c_x\in c_y \mid x\lt y;x,y\in X\}$ is also consistent; thus by completeness there is a structure $(M,\epsilon,\cdots,c_x,\cdots)$ such that * $(M,\epsilon)\models Th(V_\omega,\in)$ * $c_x\epsilon c_y$ for all $x\lt y$ in $X$

By a similar argument, (if ZFC is consistent) there are models $M'$ of classical set theory satisfying the (higher-order) property that the natural numbers object $\omega_{M'}$ of $M'$ includes your favourite total order $(X,\lt)$ \emph{as a suborder} --- of course, $M'$ isn't allowed to \emph{know} this --- notably, there is \textbf{no object} $\xi$ in $M$ such that $\{y\mid y\epsilon \xi\} = \{c_x \mid x\in X\}$.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[model theory - contents]]


\item [[definable set]], [[definable groupoid]]


\item [[type (in model theory)]]


\item [[stability in model theory]], [[geometric stability theory]]


\item [[categoricity]]


\item [[abstract model theory]]


\item [[generalized quantifier]]


\item [[nonstandard analysis]], [[set theory]], [[foundations and logic]], [[algebraic set theory]], [[forcing]], [[o-minimal structure]]


\item [[Lascar group]]


\item [[Birkhoff's HSP theorem]]


\item [[model theory and physics]]


\item entries on researchers in model theory: [[Alfred Tarski]], [[Boris Zilber]], [[Kenneth Kunen]], [[Saharon Shelah]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

Discussion of classical first-order model theory includes

\begin{itemize}%
\item Stanford Encyclopedia of Philosophy, \emph{\href{http://plato.stanford.edu/entries/modeltheory-fo/}{First-order Model Theory}}


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Model_theory}{Model theory}}


\item [[Wilfrid Hodges]], \emph{Model Theory}, Cambridge University Press 1993; \emph{A shorter model theory}, Cambridge UP 1997



\end{itemize}
Other basic texts on model theory include

\begin{itemize}%
\item Katrin Tent, [[Martin Ziegler]], \emph{A course in model theory}, Lecture Notes in Logic, Cambridge University Press, April 2012


\item Chen Chung Chang, H. Jerome Keisler, \emph{Model Theory. Studies in Logic and the Foundations of Mathematics}. 1973, 1990, Elsevier.


\item David Marker, \emph{Model Theory: An Introduction} Graduate Texts in Mathematics \textbf{217} (2002)


\item C U Jensen, H Lenzing, \emph{Model theoretic algebra: with particular emphasis on fields, rings, modules} (1989)


\item \href{http://shelah.logic.at}{Shelah archive}, \href{http://shelah.logic.at/books.html}{Shelah's books}


\item Gerald E Sacks, \emph{Saturated model theory}, Benjamin 1972



\end{itemize}
Specific discussion of generalization of classical first-order model theory to more general kinds of [[logic]] include\ldots{}

\ldots{}to [[modal logic]]:

\begin{itemize}%
\item Valentin Goranko, Martin Otto, \emph{Model theory of modal logic}, \href{http://www.mathematik.tu-darmstadt.de/~otto/papers/mlhb.pdf}{pdf}

\end{itemize}
\ldots{}to [[infinitary logic]]:

\begin{itemize}%
\item H. Keisler. \emph{Model theory for infinitary logic}, North-Holland, Amsterdam, 1971.

\end{itemize}
Discussion of aspects of [[category theory]] and [[categorical logic]] ([[accessible categories]]) in model theory includes

\begin{itemize}%
\item [[Michael Makkai]], [[Robert Paré]], \emph{Accessible categories: the foundations of categorical model theory}, Contemporary mathematics vol 104, AMS 1989


\item [[Tibor Beke]], [[Ji?í Rosický]], \emph{Abstract elementary classes and accessible categories}, 2011 (\href{http://www.math.muni.cz/~rosicky/papers/elem7.pdf}{pdf})


\item B.Hart et. al (eds.), \emph{Models, Logics, and Higher-dimensional Categories, A Tribute to the Work of Mihali Makkai}, AMS (2011)



\end{itemize}
Discussion of [[geometric stability theory]] includes

\begin{itemize}%
\item Anand Pillay, \emph{Geometric stability theory}


\item [[Boris Zilber]], \emph{Elements of geometric stability theory}, lecture notes, \href{http://people.maths.ox.ac.uk/zilber/est.pdf}{pdf}; \emph{On model theory, non-commutative geometry and physics}, (survey draft) \href{http://people.maths.ox.ac.uk/zilber/bul-survey.pdf}{pdf}; \emph{Zariski geometries}, book, \href{http://people.maths.ox.ac.uk/zilber/s.pdf}{draft pdf}; On model theory, noncommutative geometry and physics, conference talk, \href{http://zomobo.net/play.php?id=wnhbpIQBpiE}{video}


\item John T. Baldwin, \emph{Fundamentals of stability theory}



\end{itemize}
Discussion of [[motivic integration]] includes

\begin{itemize}%
\item [[David Kazhdan]], \emph{Lecture notes in motivic integration}, with intro to logic and model theory, \href{http://www.ma.huji.ac.il/~kazhdan/Notes/motivic/b.pdf}{pdf}


\item R. Cluckers, J. Nicaise, J. Sebag (Editors), \emph{Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry}, 2 vols. London Mathematical Society Lecture Note Series \textbf{383}, \textbf{384}



\end{itemize}


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