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\begin{document}

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\section*{elimination of quantifiers}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{semantic_characterizations_of_qe}{Semantic characterizations of QE}\dotfill \pageref*{semantic_characterizations_of_qe} \linebreak
\noindent\hyperlink{qe_extends_to_coslice_categories}{QE extends to co-slice categories}\dotfill \pageref*{qe_extends_to_coslice_categories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

One thinks of [[existential quantifier | existential quantification]] as \emph{[[projection]]}. A theory eliminates quantifiers when ``the projection of a primitive (quantifier-free) definable set is still primitive (quantifier-free).''

For example, suppose we're working in the theory of a [[vector space]] over a fixed field $\mathbb{F}$. Let $A$ be a linear transformation $\mathbb{F}^{n+1} \to \mathbb{F}^m$. Let $\vec{b}$ be an $m$-tuple, $x$ a variable, and $\vec{y}$ an $n$-tuple of variables. Then the solution set for

\begin{displaymath}
\exists x A(x, \vec{y}) = \vec{b}
\end{displaymath}
is the projection of the solution set for

\begin{displaymath}
A(x, \vec{y}) = \vec{b}
\end{displaymath}
to the $\vec{y}$-hyperplane. Since the projection of a (translated) hyperplane is again a (translated) hyperplane, there exists some $B : \mathbb{F}^n \to \mathbb{F}^m$ such that $B(\vec{y}) = \vec{b} \iff (\exists x) \left(A(x, \vec{y}) = \vec{b}\right).$

This is an instance of the fact that the theories of [[vector space | vector spaces]] over division rings admits quantifier elimination. It's nice when this happens, and when it doesn't, one asks for minimal ways to expand the language to make it happen. That [[ACF]] eliminates quantifiers is a special case of [[Chevalley's theorem on constructible sets]] (``the image of a constructible set is constructible'').

Often, in order to prove quantifier-elimination, one develops an \emph{effective} procedure for reducing an arbitrary sentence to a quantifier-free one, which shows that the theory is [[decidability|decidable]]; this is how Tarski proved that the theory [[RCF]] of real closed fields is decidable.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An $\mathcal{L}$-theory $T$ \textbf{eliminates quantifiers} if every $\mathcal{L}$-formula $\varphi(x)$ is equivalent to a quantifier-free $\mathcal{L}$-formula $\psi(x)$, i.e.

\begin{displaymath}
T \models \varphi(x) \leftrightarrow \psi(x).
\end{displaymath}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item Any theory is [[interpretation|bi-interpretable]] with an expanded theory where every definable set is named with a predicate, called its [[coherent logic|Morleyization]], which eliminates quantifiers.


\item Therefore, whether or not a structure has quantifier elimination depends on the language being used to describe the structure.


\item Quantifier elimination implies [[model complete theory | model completeness]]: all submodels are elementary submodels and all models are [[existentially closed theory | existentially closed]].


\item It is an informative exercise to try to prove QE syntactically (a ``tractable'' case like [[DLO]] suffices) to appreciate the usefulness of the semantic/model-theoretic characterization of QE below.


\item The characterizations below (and in general important tools in model theory like the [[compactness theorem]], the [[Lowenheim-Skolem theorem]], etc.) only work for \emph{infinite} models.


\item Since quantifier elimination is equivalent to [[substructure complete theory\#substructureconverse|substructure completeness]] and [[elementary equivalence|elementarily equivalent]] finite structures are isomorphic, (the theory of a) finite [[first-order structure]] eliminates quantifiers if and only if it is [[Fraisse limit|ultrahomogeneous]].



\end{itemize}
\hypertarget{semantic_characterizations_of_qe}{}\subsection*{{Semantic characterizations of QE}}\label{semantic_characterizations_of_qe}

\begin{prop}
\label{}\hypertarget{}{}
A first-order theory $T$ eliminates quantifiers if and only if it is substructure-complete: $T$ plus the [[quantifier-free diagram]] of any substructure of any model of $T$ form a complete theory.

\end{prop}
\begin{proof}
See [[substructure complete theory\#QEiffsubstructurecomplete|substructure completeness]].

\end{proof}
\begin{theorem}
\label{QEcriterion}\hypertarget{QEcriterion}{}
Let $T$ be a theory. The following are equivalent:

\begin{enumerate}%
\item $T$ eliminates quantifiers.


\item For every $M$ and $N$ models of $T$ such that $N$ is $|M|^+$-[[type (in model theory)|saturated]] and every proper substructure $A \subseteq M$ such that there is an embedding $i \colon A \to N$, there exists a proper intermediate substructure $A'$, $A \subsetneq A' \subseteq M$ and an extension $i' \colon A' \to N$ of $i$ to $A'$.



\end{enumerate}
\end{theorem}
\begin{proof}
(1. $\implies$ 2.) Suppose $T$ eliminates quantifiers. Since this is equivalent to [[substructure-completeness]], choose $x \in M \backslash A$, and let $B \overset{df}{=} \langle A \cup \{x\} \rangle$ be the substructure of $M$ generated by $A \cup \{x\}.$ Then, since quantifier-free statements are transferred by embeddings, the [[type (in model theory)]] of $x$ over $A$ in $M$ is finitely realized in $N$ and therefore realized via $|M|^+$-saturation by some $x'$.

A standard [[back-and-forth argument]] starting at $x'$ lets us build our required extension.

(2. $\implies$ 1.) By possibly transfinitely iterating 2., we obtain an embedding $M \to N.$ As we can always embed a $T$-model inside a larger, arbitrarily saturated $T$-model, we therefore have [[model completeness|amalgamation]].

This gives [[model completeness]]: since model completeness is equivalent to every $T$-submodel being an [[elementary embedding|elementary submodel]], it suffices by the Tarski-Vaught test (and an induction on complexity of formulas) to test that whenever $m$ is a tuple from $M$, $\varphi(x,y)$ is a quantifier-free formula, and $N \models \exists x \varphi(x, m)$, then $M \models \exists x \varphi(x,m)$ (i.e. that $M$ is [[existentially closed]].)

Embed $M$ into a sufficiently saturated elementary extension $^*M$ (by taking a large enough [[ultrapower]], if you like). Then by (2.), the elementary embedding $M \preceq ^*M$ extends along the embedding $M \hookrightarrow N$ to an embedding $N \hookrightarrow ^*M$. If $N \models \exists x \varphi(x,m)$, then there is a witness $n \in N$ such that $N \models \varphi(n,m)$. Since $\varphi$ was quantifier-free, this transfers along the embedding $N \hookrightarrow ^*M$, so that $^*M \models \varphi(n,m)$, and therefore $^*M \models \exists x \varphi(x,m).$ This then transfers down the elementary embedding $M \preceq ^*M$, so $M \models \exists x \varphi(x,m).$ Since $M$ and $N$ were arbitrary, $T$ is model complete.

Since [[model completeness\#substructureconverse|model completeness + amalgamation implies substructure completeness]] and [[substructure complete theory\#QEiffsubstructurecomplete|substructure completeness is equivalent to quantifier elimination]], the theorem is proved.

\end{proof}
\begin{cor}
\label{}\hypertarget{}{}
[[ACF]] admits quantifier elimination. Furthermore, the theories $\mathsf{ACF}_p$ (where $p$ is specification of the characteristic) are complete.

\end{cor}
\begin{proof}
We put ourselves into the situation of the above theorem. Let $E$ and $F$ be algebraically closed fields such that $F$ is $|E|^+$-saturated. Let $R$ be a subring of $E$ with $i : R \to F$ an embedding. As $E$ is a field, $R$ is an integral domain, and thus $E$ contains the fraction field $\operatorname{Frac}(R)$ of $R$. The embedding $i$ extends uniquely to $\operatorname{Frac}(R)$. By a back-and-forth argument, $i$ extends uniquely to $\operatorname{Frac}(R)^{\operatorname{alg}}$. Therefore, we can assume that $R$ is algebraically closed. Let $a \in E \backslash R$. Then $a$ is transcendental over $R$, and by saturation its [[type (in model theory) | type]] is realized by some $b \in F$ which is transcendental over the image $i(R)$ of $R$ under $i$. Then $a \mapsto b$ induces an embedding $R[a] \overset{i'}{\to} i(R)[b]$, which extends $i$. Thus the hypotheses of the theorem are satisfied and [[ACF]] admits QE.

That $\mathsf{ACF}$ becomes complete after specifying a characteristic follows from the fact that in any theory with quantifier elimination, naming a substructure (thus passing to the theory of all models which contain a copy of that substructure) makes the theory complete, and every field of characteristic $p$ contains the prime field of characteristic $p$.

\end{proof}
\hypertarget{qe_extends_to_coslice_categories}{}\subsection*{{QE extends to co-slice categories}}\label{qe_extends_to_coslice_categories}

Let $T$ eliminate quantifiers. Let $K$ be a substructure (not necessarily a submodel) of some model $A$ of $T$. Then the theory $T/K$ of models of $T$ interpreting a copy of $K$ also has quantifier elimination.

In particular, if we specialize to [[ACF]], this tells us that not only do all algebraically closed fields admit quantifier elimination in the language of rings, but so do all algebraically closed fields containing a fixed ring $R$ (i.e. the geometric points of $\operatorname{Spec}(R)$), in the language of rings with constants for the elements of $R$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[quantification]]


\item [[quantifier-free diagram]]


\item [[decidability]]



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

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Quantifier_elimination}{quantifier elimination}
\item Wilfrid Hodges, \emph{Model Theory}, Cambridge University Press 1993

\end{itemize}
category: logic, model theory [[!redirects quantifier elimination]] [[!redirects quantifier-elimination]]



\end{document}