Restricted fragments of class-sized logics regain compactness and Löwenheim-Skolem properties that exactly characterize Weak Vopěnka's Principle, Ord is Woodin, and Shelah cardinals.
Inner Models and Large Cardinals , volume 5 of de Gruyter Series in Logic and its Applications
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Model theory of class-sized logics
Restricted fragments of class-sized logics regain compactness and Löwenheim-Skolem properties that exactly characterize Weak Vopěnka's Principle, Ord is Woodin, and Shelah cardinals.