We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate ($\models^*$) appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension $\Theta$ of ZF there is a finitely axiomatizable extension $\Theta'$ of GB that is a conservative extension of $\Theta$. We also prove a conservative extension result that justifies the use of $\models^*$ to characterize ground models for forcing constructions.
"Satisfaction relations for proper classes: applications in logic and set theory." J. Symbolic Logic 78 (2) 345 - 368, June 2013. https://doi.org/10.2178/jsl.7802010