304 research outputs found
The modal logic of forcing
What are the most general principles in set theory relating forceability and
truth? As with Solovay's celebrated analysis of provability, both this question
and its answer are naturally formulated with modal logic. We aim to do for
forceability what Solovay did for provability. A set theoretical assertion psi
is forceable or possible, if psi holds in some forcing extension, and
necessary, if psi holds in all forcing extensions. In this forcing
interpretation of modal logic, we establish that if ZFC is consistent, then the
ZFC-provable principles of forcing are exactly those in the modal theory known
as S4.2.Comment: 31 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
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