We study various classes of maximality principles, MP(κ,Γ), introduced by J.D. Hamkins in [6], where Γ defines a class of forcing posets and κ is an infinite cardinal. We explore the consistency strength and the relationship of MP(κ,Γ) with various forcing axioms when κ ∈ {ω, ω1}. In particular, we give a characterization of bounded forcing axioms for a class of forcings Γ in terms of maximalit...

On Subcomplete Forcing by Kaethe Lynn Bruesselbach Minden Adviser: Professor Gunter Fuchs I survey an array of topics in set theory and their interaction with, or in the context of, a novel class of forcing notions: subcomplete forcing. Subcomplete forcing notions satisfy some desirable qualities; for example they don’t add any new reals to the model, and they admit an iteration theorem. While ...

In this paper we consider Foreman’s maximality principle, which says that any non-trivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We observe that it is consistent that every c.c.c. forcing adds a real and that for every uncountable regular cardinal κ, every κ-closed forcing of size 2<κ collapses some cardinal.

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension V P and all subsequent extensions V P∗Q̇ holds already in V . It follows, in fact, that such sentences must also hold in all forcing extensions of V . In modal terms, therefore, the Maximality Principle is exp...

The Necessary Maximality Principle for c.c.c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. The Necessary Maximality Principle for c.c.c. forcing, denoted 2mpccc(R), asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already tr...

The Necessary Maximality Principle for c.c.c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. The Necessary Maximality Principle for c.c.c. forcing, denoted 2mpccc(R), asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already tr...

Abstract It is shown that the resurrection axiom and maximality principle may be consistently combined for various iterable forcing classes. The extent to which overlap explored via local principle.

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. T...