# Abstract # The operation $\Delta$ Let $\mathfrak{L}^{\mathrm{W}}$ be [[Second order logic#$ mathcal{L} { mathrm{W}}$ - Weak second order logic|weak second order logic]]. Let HYP be the smallest admissible set containing $\omega$ and let $\mathcal{L}_{\text {HYP }}$ be the admissible fragment of $\mathfrak{L}_{\infty \omega}$ given by HYP, except that only formulas with a finite number of symbols are considered. **4.1. Theorem**. $\Delta(\mathcal{L}^{\mathrm{W}}) \equiv \Delta\left(\mathcal{L}\left(\mathrm{Q}_0\right)\right) \equiv \mathcal{L}_{\text {HYP}}$.