  The founding of the theory of cylindric algebras, by Alfred Tarski, was a
conscious effort to create algebras out of first order predicate calculus. Let
$n\in\omega$. The classes of non-commutative cylindric algebras ($NCA_n$) and
weakened cylindric algebras ($WCA_n$) were shown, by Istv\'an N\'emeti, to be
examples of decidable versions of first order logic with $n$ variables. In this
article, we give new proofs for the decidability of the equational theories of
these classes. We also give an answer to the open problem, posed by N\'emeti in
1985, addressing the atomicity of the finitely generated free algebras of these
classes. We prove that all the finitely generated free algebras of the
varieties $NCA_n$ and $WCA_n$ are not atomic. In other words, we prove that the
corresponding versions of first order logic have weak G\"odel's incompleteness
property.
