George Joseph Leibman earned his Ph.D. under my supervision in June, 2004 at the CUNY Graduate Center. He was my first Ph.D. student. Being very interested both in forcing and in modal logic, it was natural for him to throw himself into the emerging developments at the common boundary of these topics.  He worked specifically on the natural extensions of the maximality principle where when one considers a fixed definable class $\mathrm{\Gamma }$ of forcing notions.  This research engaged with fundamental questions about the connection between the forcing-theoretic properties of the forcing class $\mathrm{\Gamma }$ and the modal logic of its forcing validities, and was a precursor of later work, including joint work, on the modal logic of forcing.

George Leibman

web page | math genealogy | MathSciNet | ar$\chi$iv | related posts

George Leibman, “Consistency Strengths of Modified Maximality Principles,” Ph.D. thesis, CUNY Graduate Center, 2004.  ar$\chi$iv

Abstract. The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension $V^{\mathbb{P}}$, and remains true in any further forcing extension of $V^{\mathbb{P}}$, then it is true in all forcing extensions of $V$.  A modified maximality principle ${\text{MP}}_{\mathrm{\Gamma }}$ arises when considering forcing with a particular class $\mathrm{\Gamma }$ of forcing notions. A parametrized form of such a principle, ${\text{MP}}_{\mathrm{\Gamma }}(X)$, considers formulas taking parameters; to avoid inconsistency such parameters must be restricted to a specific set $X$ which depends on the forcing class $\mathrm{\Gamma }$ being considered. A stronger necessary form of such a principle, $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.1.0/svg/25fb.svg">}{\text{MP}}_{\mathrm{\Gamma }}(X)$, occurs when it continues to be true in all $\mathrm{\Gamma }$ forcing extensions.

This study uses iterated forcing, modal logic, and other techniques to establish consistency strengths for various modified maximality principles restricted to various forcing classes, including ccc, COHEN, COLL (the forcing notions that collapse ordinals to $\omega$), $<\kappa$ directed closed forcing notions, etc., both with and without parameter sets. Necessary forms of these principles are also considered.