286,624 research outputs found
Non-elementary proper forcing
We introduce a simplified framework for ord-transitive models and Shelah's
non elementary proper (nep) theory. We also introduce a new construction for
the countable support nep iteration
Preserving levels of projective determinacy by tree forcings
We prove that various classical tree forcings -- for instance Sacks forcing,
Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve
the statement that every real has a sharp and hence analytic determinacy. We
then lift this result via methods of inner model theory to obtain
level-by-level preservation of projective determinacy (PD). Assuming PD, we
further prove that projective generic absoluteness holds and no new equivalence
classes classes are added to thin projective transitive relations by these
forcings.Comment: 3 figure
Dependent choice, properness, and generic absoluteness
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory
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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