From [[Barwise, Feferman (eds.) - Model-theoretic logics]] chapter II section 3 >[!info] Definition > Let $\mathcal{L}$ be a logic and $\mathfrak{R}$ a class of $τ$-structures. >- We say that $\mathfrak{R}$ is an *elementary class* in $\mathcal{L}$ (or that $\mathfrak{R}$ is EC in $\mathcal{L}$, or that $\mathfrak{R}\in \mathrm{EC}_{\mathcal{L}}$ ) iff there is $φ\in \mathcal{L}[\tau]$ such that $\mathfrak{R}=\mathrm{Mod}^{\tau}_{\mathcal{L}}(\varphi)$. >- We say that $\mathfrak{R}$ is a *projective class* in $\mathcal{L}$ (or that $\mathfrak{R}$ is PC in $\mathcal{L}$ or that $\mathfrak{R}\in \mathrm{PC}_{\mathcal{L}}$ iff > - *One sorted:* there is $τ' \supseteq τ$, and a class $\mathfrak{R} of $τ'$-structures, $\mathfrak{R}\in \mathrm{EC}_{\mathcal{L}}$, such that $\mathfrak{R}=\{ \mathfrak{A}\restriction\tau \mid \mathfrak{A}\in \mathfrak{R}'\}$. > - *Many-sorted:* same but require that $\tau'$ has the same sorts as $\tau$. >- We say that $\mathfrak{R}$ is a *relativized projective class* in $\mathcal{L}$ (or that $\mathfrak{R}$ is RPC in $\mathcal{L}$ or that $\mathfrak{R}\in \mathrm{RPC}_{\mathcal{L}}$ iff > - *One sorted:* there is $τ' \supseteqτ$, a unary realtion symbol $U\in \tau' \smallsetminus \tau$, and a class $\mathfrak{R}'$ of $τ'$-structures, $\mathfrak{R}\in \mathrm{EC}_{\mathcal{L}}$, such that $\mathfrak{R}=\{ (\mathfrak{A}\restriction\tau)|U^{\mathfrak{A}} \mid \mathfrak{A}\in \mathfrak{R}, U^{\mathfrak{A}}\text{ is } \tau \text{-closed in }\mathfrak{A} \}$. > - *Many sorted:* there is $τ' \supseteq τ$, and a class $\mathfrak{R}'$ of $τ'$-structures, $\mathfrak{R}'\in \mathrm{EC}_{\mathcal{L}}$, such that $\mathfrak{R}=\{ \mathfrak{A}\restriction\tau \mid \mathfrak{A}\in \mathfrak{R}'\}$. > - PC in $\mathcal{L}$ classes are also refered to as $\Sigma_{1}^{1}(\mathcal{L})$ >- RPC in $\mathcal{L}$ classes are also refered to as $\Sigma(\mathcal{L})$ >- We say that $\mathfrak{R}$ is *$\Delta^{1}_{1}$ in $\mathcal{L}$* (or that $\mathfrak{R}$ is $\Delta^{1}_{1}(\mathcal{L})$-definable, or that $\mathfrak{R} \in \Delta^{1}_{1}(\mathcal{L})$) if both $\mathfrak{R}$ and its complement $\bar{\mathfrak{R}}$ are PC in $\mathcal{L}$ >- We say that $\mathfrak{R}$ is *$\Delta$ in $\mathcal{L}$* (or that $\mathfrak{R}$ is $\Delta(\mathcal{L})$-definable, or that $\mathfrak{R} \in \Delta(\mathcal{L})$) if both $\mathfrak{R}$ and its complement $\bar{\mathfrak{R}}$ are RPC in $\mathcal{L}$ >[!note] Remark >Then notion of $\Delta^{1}_{1}$ dosen't appear in [[Barwise, Feferman (eds.) - Model-theoretic logics]] , but in [[Väänänen - Δ-Extension and Hanf-numbers]], where the terminology is different: only the many sorted case is considered, relativized projective classes are called *projective classes* and projective classes are called *simple projective classes*.