\cite{calculi-orig} proves some results about undecidable problems for propositional calculi.
In the preliminaries it introduces notations and definitions from scratch:
``Let us consider the language consisting of an infinite set of propositional variables
\begin{labeledquote}
Let us consider the language consisting of an infinite set of propositional variables
$V$ and the signature $\Sigma$, i.e., a finite set of connectives. Letters
$x, y, z, u, p$ , etc., are used to denote propositional variables.
Usually connectives are binary or unary such as $\neg, \vee, \wedge,$ or $\arr$.''
Usually connectives are binary or unary such as $\neg, \vee, \wedge,$ or $\arr$.
\end{labeledquote}
However, it still relies on the literature to fully define the local concepts:
``Propositional formulas or $\Sigma$-formulas are built up from the signature
$\Sigma$ and propositional variables $V$\emph{in the usual way}. '' [emphasis ours]
The paper continues by introducing previous results, by mapping them (implicitly) to the definitions introduced
locally. For instance :
\begin{labeledquote}
\begin{theorem}[Theorem 2.1]
(Linial and Post, 1949). \textbf{Axm}, \textbf{Ext}, and \textbf{Cmpl} are undecidable for $\textbf{Cl}_{\{\neg, \wedge\}}$.
\end{theorem}
\end{labeledquote}
where \textbf{Axm}, \textbf{Axm}, \textbf{Ext}, \textbf{Cmpl} and \textbf{Cl} are all defined locally.
Having established local concepts and their properties (via the literature) the paper continues with its contribution.
...
...
@@ -59,6 +66,7 @@ Having established local concepts and their properties (via the literature) the
\cite{covers-orig} discusses covers of the multiplicative group of an algebraically closed field which are formally defined in the beginning
of the paper as follows:
\begin{labeledquote}
\begin{definition}
Let $V$ be a vector space over $Q$ and let $F$ be an algebraically closed field
of characteristic $0$. A \emph{cover of the multiplicative group} of $F$ is a structure represented by
...
...
@@ -67,6 +75,7 @@ $0 \arr K \arr V \arr F^∗ \arr 1$ ,
where the map $V \arr F^*$ is a surjective group homomorphism from $(V, +)$ onto
$(F^∗, \cdot)$ with kernel $K$. We will call this map \emph{exp}.
\end{definition}
\end{labeledquote}
When, establishing results, \cite{covers-orig} mentions ``Moreover, with an additional axiom (in $L_{\omega_1\omega}$) stating $K \cong Z$, the class is categorical
in uncountable cardinalities. This was originally proved in \cite{covers-13} but an error was later found
...
...
@@ -82,11 +91,12 @@ extends the result in \cite{covers-13} and additionally fixes a hole in the proo