7,273 research outputs found
Set mapping reflection
In this note we will discuss a new reflection principle which follows from
the Proper Forcing Axiom. The immediate purpose will be to prove that the
bounded form of the Proper Forcing Axiom implies both that 2^omega = omega_2
and that L(P(omega_1)) satisfies the Axiom of Choice. It will also be
demonstrated that this reflection principle implies that combinatorial
principle Square(kappa) fails for all regular kappa > omega_1.Comment: 11 page
Definable MAD families and forcing axioms
We show that under the Bounded Proper Forcing Axiom and an anti-large
cardinal assumption, there is a MAD family.Comment: 13 page
The bounded proper forcing axiom and well orderings of the reals
We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(Ļ_1) which is Ī_1 definable with parameter a subset of Ļ_1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes N_2 and also satisfies BPFA must contain all subsets of Ļ_1. We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the HƤrtig quantifier is not lightface projective
Boolean Valued Models, Saturation, Forcing Axioms
This dissertation will focus on Boolean-valued models, giving some insight into the theory of Boolean ultrapowers, and developing the connection with forcing axioms and absoluteness results. This study will be divided into three chapters.
The first chapter provides the basic material to understand the subsequent work.
Boolean-valued models are well known in set theory for independence results and the development of forcing. In the second chapter of this dissertation, Boolean-valued models are studied from a general point of view. In particular, we give the definition of B-valued model for an arbitrary first-order signature, and we study Boolean ultrapowers as a general model-theoretic technique.
A more ambitious third chapter develops the connection with forcing axioms and absoluteness results. From a philosophical point of view, forcing axioms are very appealing. Not only do they imply that the Continuum Hypothesis is false, but also they are particularly successful in deciding many independent statements in mathematics. First, we give an interesting formulation of bounded forcing axioms in terms of absoluteness. Furthermore, we prove that the Axiom of Choice is a āglobalā forcing axiom, and that also some large cardinal axioms are in fact natural generalizations of forcing axioms
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