Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
$\begingroup$Sure. Just check all the possible interpretations for all the formula's function and relation symbols over the domain $\{1,2,\dots,k\}$. $\endgroup$
$\begingroup$Note that in Andreas's answer (and the question, too), $k$ can be replaced by $k$ or fewer elements without changing the character of decidability. Gerhard "Ask Me About System Design" Paseman, 2012.11.19$\endgroup$
$\{1,2,\dots,k\}$
. $\endgroup$