Maximality, Recurrence, Ground

The Maximality Principle (introduced by Hamkins) asserts that any forceably necessary statement is true. The Recurrence Axiom (introduced by Fuchino and Usuba) asserts that any forceable statement is true in some ground, where an inner model W is said to be a ground if VV is a set-sized forcing extension of W. In this talk, we will explore natural variants of these principles by restricting the complexity of statements, allowing parameters, and varying the class of forcing posets. For example, we discuss the (in)compatibility of these variants with the Ground Axiom (introduced by Hamkins and Reitz), which asserts that there are no non-trivial grounds. This talk is based on joint work with Sakaé Fuchino and Francesco Parente.