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\begin{document}

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\section*{completeness theorem}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak


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

A formal system is \emph{complete} or \emph{semantically complete} when all its [[tautologies]] are [[theorems]]. Since tautologies are defined by reference to ``all models'', this can depend on the class of models under consideration.

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

\begin{itemize}%
\item [[Kurt Gödel]] showed that [[classical mathematics|classical]] [[first-order logic]] is complete, that is, any first-order sentence, $\phi$, which is true under any interpretation, is provable from the axioms of first-order logic. This is generally known as \textbf{the completeness theorem}. Here the ``interpretations'' are the models in [[sets]], assigning a set to every sort and a binary [[truth value]] $\{\bot,\top\}$ to every proposition.


\item If in an [[infinitary logic]], infinitely many quantifiers may appear in sentences, then completeness does not hold.


\item In [[constructive mathematics]], G\"o{}del's proof is not available, and (perhaps depending on definitions) constructive logic may not be complete with respect to ordinary models in sets. However, there are still completeness theorems with respect to models in arbitrary [[categories]], which generally proceed by way of construction of a [[syntactic category|syntactic]] model in which the true statements are precisely the theorems.



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

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


\item [[incompleteness theorem]]



\end{itemize}
[[!redirects Gödel-Henkin completeness theorem]] [[!redirects Gödel completeness theorem]]



\end{document}