Inner models from extended logics and the Delta-operation

Abstract

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 Gödel'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 $Δ$-extension $Δ(\mathcal{L})$ 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 $C(Δ(\mathcal{L}))$, and in one case we show this follows from the existence of $0^{\sharp}$.

Theorems & Definitions (31)

• Definition 1
• Definition 2
• Definition 6
• Definition 9
• Definition 11
• Proposition 13
• Theorem 14
• proof
• Theorem 15
• Theorem 16