A Simple Maximality Principle

Joel David Hamkins

The Maximality Principle I propose, following an idea of Christophe Chalons, asserts that any sentence that holds in some forcing extension V^{P} and all subsequent extensions V^{P ∗Q}, holds already in V and all forcing extensions of V. The principle is naturally expressed by the modal logic axiom ’(♢□ϕ) implies □ϕ)’. In this talk I will prove the relative consistency of the Maximality Principle and analyze its relative consistency strength, showing it to be closely related to the scheme asserting that V_{δ} is an elementary substructure of V.

Address: The City University of New York, CUNY Graduate Center, Mathematics Program, 365 Fifth Avenue, New York, NY 10016, USA

E-mail: jhamkins@gc.cuny.edu