Given an uncountable regular cardinal , we study the structural properties of the class of all sets of functions from to that are definable over the structure by a -formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of together with large cardinal axioms. In this paper, we present other canonical extensions of that provide a strong structure theory for these classes. These axioms are variations of the Maximality Principle introduced by Stavi and Väänänen and later rediscovered by Hamkins.
"Closed Maximality Principles and Generalized Baire Spaces." Notre Dame J. Formal Logic 60 (2) 253 - 282, May 2019. https://doi.org/10.1215/00294527-2019-0004