If $\mathcal{ L}$ is an abstract logic (a.k.a. model theoretic logic), we can define the inner model $C(\mathcal{ L})$ by replacing first order logic with $\mathcal{ L}$ in G\"odel's definition of the inner model $L$ of constructible sets. Set theoretic properties of such inner models $C(\mathcal{ L})$ have been investigated recently and a spectrum of new inner models is emerging between $L$ and $\mathrm{ HOD}$. The topic of this paper is the effect on $C(\mathcal{ L})$ of a slight modification of $\mathcal{ L}$ i.e. how sensitive is $C(\mathcal{ L})$ on the exact definition of $\mathcal{ L}$? The $\Delta$-extension $\Delta(\mathcal{ L})$ of a logic is generally considered a "mild" extension of $\mathcal{ L}$. We give examples of logics $\mathcal{ L}$ for which the inner model $C(\mathcal{ L})$ is consistently strictly smaller than the inner model $C(\Delta(\mathcal{ L}))$, and in one case we show this follows from the existence of $0^{ \sharp}$.
Categories: math.LO