scholarly journals Category forcings, ${MM}^{+++}$, and generic absoluteness for the theory of strong forcing axioms

2015 ◽  
Vol 29 (3) ◽  
pp. 675-728 ◽  
Author(s):  
Matteo Viale
Keyword(s):  
Forcing Axioms ◽  
Generic Absoluteness

scholarly journals SUBCOMPLETE FORCING, TREES, AND GENERIC ABSOLUTENESS

2018 ◽  
Vol 83 (3) ◽  
pp. 1282-1305 ◽  
Author(s):  
GUNTER FUCHS ◽  
KAETHE MINDEN
Keyword(s):  
Forcing Axioms ◽  
First Order ◽  
Generic Absoluteness ◽  
Subcomplete Forcing ◽  
Image Position ◽  
Aronszajn Trees ◽  
Order Structures ◽  
Forcing Axiom

AbstractWe investigate properties of trees of height ω1 and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an ω1-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcing. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width ω1. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width ω1 and generic absoluteness of ${\rm{\Sigma }}_1^1$-statements over first order structures of size ω1, also for other canonical classes of forcing.

scholarly journals Absoluteness via resurrection

2017 ◽  
Vol 17 (02) ◽  
pp. 1750005 ◽  
Author(s):  
Giorgio Audrito ◽  
Matteo Viale
Keyword(s):  
Set Theory ◽  
Order Theory ◽  
Consistency Strength ◽  
Forcing Axioms ◽  
Supercompact Cardinals ◽  
First Order ◽  
Mahlo Cardinal ◽  
Generic Absoluteness ◽  
First Order Theory ◽  
Stationary Set

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms [Formula: see text] for a class of forcings [Formula: see text] and a given ordinal [Formula: see text]), and show that [Formula: see text] implies generic absoluteness for the first-order theory of [Formula: see text] with respect to forcings in [Formula: see text] preserving the axiom, where [Formula: see text] is a cardinal which depends on [Formula: see text] ([Formula: see text] if [Formula: see text] is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for [Formula: see text] with respect to [Formula: see text] and for [Formula: see text] with respect to [Formula: see text] with [Formula: see text] is in principle possible, and we present several natural models of the Morse–Kelley set theory where this phenomenon occurs (even for all [Formula: see text] simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.

Generic Σ31 absoluteness

2004 ◽  
Vol 69 (1) ◽  
pp. 73-80 ◽  
Author(s):  
Sy D. Friedman
Keyword(s):  
Regular Cardinal ◽  
Forcing Axioms ◽  
Proper Forcing ◽  
Class Forcing ◽  
The Universe ◽  
Generic Extensions

In this article we study the strength of absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [3]. (In particular, see Theorem 3 below.) We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA.The statement “ absoluteness holds for ccc forcing” means that if a formula with real parameters has a solution in a ccc set-forcing extension of the universe V, then it already has a solution in V. The analogous definition applies when ccc is replaced by other set-forcing notions, or by class-forcing.Theorem 1. [1] absoluteness for ccc has no strength; i.e., if ZFC is consistent then so is ZFC + absoluteness for ccc.The following results concerning (arbitrary) set-forcing and class-forcing can be found in [3].Theorem 2 (Feng-Magidor-Woodin). (a) absoluteness for arbitrary set-forcing is equiconsistent with the existence of a reflecting cardinal, i.e., a regular cardinal κ such that H(κ) is ∑2-elementary in V.(b) absoluteness for class-forcing is inconsistent.We consider next the following set-forcing notions, which lie strictly between ccc and arbitrary set-forcing: proper, semiproper, stationary-preserving and ω1-preserving. We refer the reader to [8] for the definitions of these forcing notions.Using a variant of an argument due to Goldstern-Shelah (see [6]), we show the following. This result corrects Theorem 2 of [3] (whose proof only shows that if absoluteness holds in a certain proper forcing extension, then in L either ω1 is Mahlo or ω2 is inaccessible).

HAPPY AND MAD FAMILIES INL(ℝ)

2018 ◽  
Vol 83 (2) ◽  
pp. 572-597 ◽  
Author(s):  
ITAY NEEMAN ◽  
ZACH NORWOOD
Keyword(s):  
Generic Absoluteness ◽  
Mad Families ◽  
Recent Theorem ◽  
Image Position ◽  
Solovay Model ◽  
Happy Family

AbstractWe prove that, in the choiceless Solovay model, every set of reals isH-Ramsey for every happy familyHthat also belongs to the Solovay model. This gives a new proof of Törnquist’s recent theorem that there are no infinite mad families in the Solovay model. We also investigate happy families and mad families under determinacy, applying a generic absoluteness result to prove that there are no infinite mad families under$A{D^ + }$.

scholarly journals Morasses, square and forcing axioms

1996 ◽  
Vol 80 (2) ◽  
pp. 139-163 ◽  
Author(s):  
Charles Morgan
Keyword(s):  
Forcing Axioms

Forcing Axioms

Set Theory ◽  
1998 ◽  
pp. 1-21
Author(s):  
Maxim R. Burke
Keyword(s):  
Forcing Axioms

HIERARCHIES OF (VIRTUAL) RESURRECTION AXIOMS

2018 ◽  
Vol 83 (1) ◽  
pp. 283-325 ◽  
Author(s):  
GUNTER FUCHS
Keyword(s):  
Forcing Axioms ◽  
Level I ◽  
Consistency Strengths ◽  
Extendible Cardinals

AbstractI analyze the hierarchies of the bounded resurrection axioms and their “virtual” versions, the virtual bounded resurrection axioms, for several classes of forcings (the emphasis being on the subcomplete forcings). I analyze these axioms in terms of implications and consistency strengths. For the virtual hierarchies, I provide level-by-level equiconsistencies with an appropriate hierarchy of virtual partially super-extendible cardinals. I show that the boldface resurrection axioms for subcomplete or countably closed forcing imply the failure of Todorčević’s square at the appropriate level. I also establish connections between these hierarchies and the hierarchies of bounded and weak bounded forcing axioms.

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