>[!info] Definition >Let $\kappa, \lambda$ be cardinals, $\kappa\geq\lambda$ >- A logic $\mathcal{L}$ is said to be *$\kappa$-compact* whenever given a set of $\mathcal{L}$-sentences $\Sigma$ of cardinality $\kappa$, if each finite subset of $\Sigma$ has a model, then $\Sigma$ has a model. >- A logic is *(fully) compact* if it is $\kappa$-compact for every $\kappa$. >- A logic $\mathcal{L}$ is said to be *$(\kappa, \lambda)$-compact* whenever given a set of $\mathcal{L}$-sentences $\Sigma$ of cardinality $\kappa$, if each subset of $\Sigma$ of cardinality lt;\lambda$ has a model, then $\Sigma$ has a model. > >So a logic is $\kappa$-compact iff it is $(\kappa,\aleph_{0})$-compact