# $L_{\mathrm{HYP}}$ Let $\mathrm{HYP}$ be the smallest [[admissible set]] containing $\omega$ as an element, and let $\mathcal{L}_{\mathrm{HYP}}=\mathcal{L}_{\infty\omega} \cap \mathrm{HYP}$ - the admissible fragment of $\mathcal{L}_{\infty \omega}$ given by $\mathrm{HYP}$, except that only formulas with a finite number of symbols are considered. Alternatively $\mathcal{L}_{\mathrm{HYP}}$ is the smallest admissible fragment of $\mathcal{L}_{\omega_1\omega}$ More generally, $\mathrm{HYP}(x)$ denotes the smallest admissible set containing $x$. Correspondingly define $\mathcal{L}_{\mathrm{HYP}(x)}$. ## Properties See [[Barwise - Axioms for abstract model theory]] section 4. - $\mathcal{L}_{\mathrm{HYP}}$ has the interpolation property. - $\Delta(\mathcal{L}^{\mathrm{w}}) \equiv \Delta\left(\mathcal{L}\left(\mathrm{Q}_0\right)\right) \equiv \mathcal{L}_{\mathrm{HYP}}$.