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\begin{document}

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\section*{infinitary logic}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \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}

An \textbf{infinitary logic} is a logic that allows infinitely long statements or infinitely long proofs, for example, by allowing conjunctions, disjunctions and quantifier sequences, to be of infinite length. The price to pay for adopting many of these more expressive logics is the failure of [[completeness theorem|completeness]] or of [[compactness theorem|compactness]].

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

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


\item [[first-order logic]]


\item [[accessible category]]


\item [[geometric theory]]


\item [[Skolem paradox]]



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

An infinitary logic was introduced for the first time in

\begin{itemize}%
\item Ernst Zermelo, \emph{\"U{}ber Stufen der Quantifikation und die Logik des Unendlichen}, Jahresbericht der Deutschen Mathematiker-Vereinigung 41/42 (1931), pp. 568--570. (\href{http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN37721857X_0041|LOG_0038&physid=PHYS_0395}{gdz})

\end{itemize}
Other early works include Novikov and

\begin{itemize}%
\item D. A. Bochvar, \emph{\"U{}ber einen Aussagenkalk\"u{}l mit abz\"a{}hlbaren logischen Summen und Produkten}, Mathematieskii Sbornik \textbf{17} (1940), pp 65--100.

\end{itemize}
Infinitary logic has been recently studied extensively by [[Saharon Shelah]] and in categorical logic by [[Mihaly Makkai]].

\begin{itemize}%
\item [[Rami Grossberg]], \emph{Classification theory for abstract elementary classes}. Logic and Algebra, ed. Yi Zhang, Contemporary Mathematics, Vol 302, AMS, (2002), pp. 165--204, \href{http://www.math.cmu.edu/~rami/Rami-NBilgi.pdf}{pdf}


\item Rami Grossberg, Saharon Shelah, \emph{On the number of non isomorphic models of an infinitary theory which has the order property}, Part A, Journal of Symbolic Logic, 51, (1986) 302--322, \href{http://links.jstor.org/sici?sici=0022-4812%28198606%2951%3A2%3C302%3AOTNONM%3E2.0.CO%3B2-D}{jstor pdf}



\end{itemize}
[[!redirects infinitary first-order theory]]



\end{document}