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\begin{document}

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\section*{Barr&#39;s theorem}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{remark}{Remark}\dotfill \pageref*{remark} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{link}{Link}\dotfill \pageref*{link} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\textbf{Barr's theorem} was originally conjectured by [[William Lawvere]] as an infinitary generalization of the [[Deligne completeness theorem]] for [[coherent toposes]] which can be expressed as the existence of a surjection $\mathcal{S}/K\to\mathcal{E}$ for a coherent topos $\mathcal{E}$ with set of points $K$. General toposes $\mathcal{E}$ may fail to have [[enough points]] but [[Michael Barr]] showed that a surjection from a suitable [[Boolean topos]] still exists.

As surjections permit the transfer of logical properties, Barr's theorem has the following important consequence:

\begin{quote}%
If a statement in [[geometric logic]] is deducible from a [[geometric theory]] using classical [[logic]] and the [[axiom of choice]], then it is also deducible from it in [[constructive mathematics]].


\end{quote}
The proof of Barr's theorem itself, however, is highly non-constructive.

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{theorem}
\label{}\hypertarget{}{}
If $\mathcal{E}$ is a [[Grothendieck topos]], then there is a [[surjective geometric morphism]]

\begin{displaymath}
\mathcal{F} \to \mathcal{E}
\end{displaymath}
where $\mathcal{F}$ satisfies the [[axiom of choice]].

\end{theorem}
\hypertarget{remark}{}\subsection*{{Remark}}\label{remark}

[[Deligne completeness theorem|Deligne's completeness theorem]] says that a coherent Grothendieck topos has enough points in $Set$ and this corresponds to the G\"o{}del-Henkin completeness theorem for first-order theories. Similarly, Barr's theorem can interpreted as saying that a Grothendieck topos has sufficient \emph{Boolean-valued} points and is in turn closely related to Mansfield's \textbf{Boolean-valued completeness theorem} for infinitary first-order theories.

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

\begin{itemize}%
\item [[Deligne completeness theorem]]


\item [[Boolean topos]]



\end{itemize}
\hypertarget{link}{}\subsection*{{Link}}\label{link}

\begin{itemize}%
\item MO discussion on topos without points: \emph{\href{http://mathoverflow.net/questions/98729/topos-without-point-from-the-point-of-view-of-logic}{Topos Without point, from the point of view of logic.}}

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

\begin{itemize}%
\item [[Michael Barr|M. Barr]], \emph{Toposes without points} , JPAA \textbf{5} (1974) pp.265-280. (\href{http://www.math.mcgill.ca/barr/papers/top.no.pt.pdf}{preprint}) doi:\href{http://dx.doi.org/10.1016/0022-4049(74%2990037-1}{10.1016/0022-4049(74)90037-1}

\end{itemize}
Extensive discussion of the context of Barr's theorem is in chapter 7 of:

\begin{itemize}%
\item [[P. T. Johnstone]], \emph{Topos Theory} , Academic Press New York 1977 (Dover reprint 2014).

\end{itemize}
A proof sketch and a survey of its model-theoretic context is in

\begin{itemize}%
\item [[Gonzalo Reyes|Gonzalo E. Reyes]], \emph{Sheaves and concepts: A model-theoretic interpretation of Grothendieck topoi} , Cah. Top. Diff. G\'e{}o. \textbf{Cat. XVIII} no.2 (1977) pp.405-437. (\href{http://www.numdam.org/item?id=CTGDC_1977__18_2_105_0}{numdam})

\end{itemize}
For a discussion of the importance of this theorem in constructive algebra see also

\begin{itemize}%
\item [[Gavin Wraith]], \emph{Intuitionistic algebra: some recent developments in topos theory} In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pages 331--337, Helsinki, 1980. Acad. Sci. Fennica. (\href{http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0331.0338.ocr.pdf}{pdf})

\end{itemize}
For proof-theoretic approaches to Barr's theorem see

\begin{itemize}%
\item Sara Negri, \emph{Contraction-free sequent calculi for geometric theories with an application to Barr's theorem} , Archive for Mathematical Logic \textbf{42} (2003) pp.389--401.

\end{itemize}
For the connection with the Boolean-valued completeness theorem see also

\begin{itemize}%
\item R. Goldblatt, \emph{Topoi - The Categorical Analysis of Logic} , North-Holland 1982$^2$. (Dover reprint New York 2006; \href{http://projecteuclid.org/euclid.bia/1403013939}{project euclid})


\item R. Mansfield, \emph{The Completeness Theorem for Infinitary Logic} , JSL \textbf{37} no.1 (1972) pp.31-34.


\item [[Michael Makkai]], [[Gonzalo E. Reyes]], \emph{First-order Categorical Logic} , LNM \textbf{611} Springer Heidelberg 1977.



\end{itemize}


\end{document}