>[!info] Definition >Let $\kappa$ be a regular cardinal. >A logic $\mathcal{L}$ has the *Tarski property* for $\kappa$ , if the union of an L-elementary proper chain of cofinality $\geq\kappa$ is an $L$-elementary extension of all the members of the chain. >Denote this property $\mathrm{T}(\kappa)$. >$\mathcal{L}$ has the *Tarski property* if it satisfies $\mathrm{T}(\omega)$. >[!note] Remark > For first order logic this is sometimes refered to as *the union lemma*.