# The $\Delta$-extension >[!info] Definition >Let $\mathcal{L}$ be a logic. The *$\Delta$-extension of $\mathcal{L}$, $\Delta(\mathcal{L})$*, is the smallest logic $\mathcal{L}^{\ast}$ such that every [[Model classes|model class]] which is $\Delta(\mathcal{L})$ is $EC(\mathcal{L}^{\ast})$. ## Explicite constructions ### [[Barwise, Feferman (eds.) - Model-theoretic logics]] - Chapter II section 7.2 Formulas of $\Delta(\mathcal{L})$ are of the form $(\varphi_{0},\varphi_{1})$ such that $\varphi_{i} \in \Sigma(\mathcal{L})$, and their model classe are complementary ($\mathcal{M} \vDash_{\mathcal{L}}\varphi_{0} \iff \mathcal{M} \nvDash_{\mathcal{L}}\varphi_{1}$). The satisfaction is $\mathcal{M} \vDash_{\Delta(\mathcal{L})}(\varphi_{0},\varphi_{1}) \iff \mathcal{M} \vDash_{\mathcal{L}} \varphi_{0}$ ### [[Makowsky, Shelah, Stavi - Delta-Logics and generalized quantifiers]] Let L be a logic and let $\left\{K_\alpha\right\}_{\alpha \in A}$ be a list of all classes $K$ such that $K, \bar{K} \in E C_L^\tau$ for some semi-simple type $\tau$. Now put $\Delta(L)=L_{\omega \omega}\left[Q^\alpha\right]_{\alpha \in A}$ where the generalized quantifier $\mathrm{Q}^\alpha$ has $K^\alpha$ as its defining class. ## Properties [[Makowsky, Shelah, Stavi - Delta-Logics and generalized quantifiers]] - $\Delta(\mathcal{L})$ is the smallest logic for which $\Delta$-[[Definability and interpolation properties#$ Delta$-interpolation|interpolation]] holds for $\mathcal{L}$. - $\Delta(\Delta(\mathcal{L}))=\Delta(\mathcal{L})$ - If $\mathcal{L}$ is $(\kappa, \lambda)$-compact so is $\Delta(\mathcal{L})$. - If $\mathcal{L}$ is [[Unbonded logic|bounded]], so is $\Delta(\mathcal{L})$ . In fact, $\mathcal{L}$ and $\Delta(\mathcal{L})$ have the same $\mathrm{wo}$-number. - $\mathcal{L}$ and $\Delta(\mathcal{L})$ have the same [[Numbers of logics|Löwenheim]]-numbers. - $\Delta$ does not preserve the [[Back and forth|Karp property]]. - $\Delta$ does not preserve the [[Tarski property]]. ## Motivation >But the $\Delta$-closure is more than just a technical tool to construct logics. >It is a closure operator motivated by Beth's Theorem which adds to a logic $L$ everything which is, in some sense, implicit in it. >- [[Makowsky, Shelah, Stavi - Delta-Logics and generalized quantifiers]] >[!note] Remark >In a footnote, [[Barwise - Axioms for abstract model theory|Barwise]] says >>It is not clear who first considered the operation $\Delta$. We first learned of it from conversations with Friedman and Keisler in the spring of 1971. ## Resources [[Barwise, Feferman (eds.) - Model-theoretic logics]] chapter II section 7.2 [[Makowsky, Shelah, Stavi - Delta-Logics and generalized quantifiers]] [[Makowsky, Shelah - The theorems of Beth and Craig in abstract model theory. I. The abstract setting]] [[Barwise - Axioms for abstract model theory]] section 4 [[Väänänen - Δ-Extension and Hanf-numbers]]