Notes on Forcing Axioms

10.1142/9013 ◽  
2013 ◽  
Author(s):  
Stevo Todorcevic ◽  
Chitat Chong ◽  
Qi Feng ◽  
Theodore A Slaman ◽  
W Hugh Woodin ◽  
...  
Keyword(s):  
Forcing Axioms

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).

The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal

2002 ◽  
Vol 8 (1) ◽  
pp. 91
Author(s):  
Paul B. Larson ◽  
W. Hugh Woodin
Keyword(s):  
Forcing Axioms ◽  
Axiom Of Determinacy ◽  
Nonstationary Ideal

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

Forcing axioms and cardinal arithmetic

2010 ◽  
pp. 328-360
Author(s):  
Boban Veličković
Keyword(s):  
Forcing Axioms ◽  
Cardinal Arithmetic

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.

ON RESURRECTION AXIOMS

2015 ◽  
Vol 80 (2) ◽  
pp. 587-608 ◽  
Author(s):  
KONSTANTINOS TSAPROUNIS
Keyword(s):  
Lower Bounds ◽  
Forcing Axioms ◽  
Extendible Cardinals ◽  
Weak Square

AbstractThe resurrection axioms are forms of forcing axioms that were introduced recently by Hamkins and Johnstone, who developed on earlier ideas of Chalons and Veličković. In this note, we introduce a stronger form of resurrection (which we callunboundedresurrection) and show that it gives rise to families of axioms which are consistent relative to extendible cardinals, and which imply the strongest known instances of forcing axioms, such as Martin’s Maximum++. In addition, we study the unbounded resurrection postulates in terms of consistency lower bounds, obtaining, for example, failures of the weak square principle.

On a generalization of Jensen's □κ, and strategic closure of partial orders

1983 ◽  
Vol 48 (4) ◽  
pp. 1046-1052 ◽  
Author(s):  
Dan Velleman
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Recent Work ◽  
Partial Orders ◽  
Closure Condition ◽  
Forcing Axioms ◽  
Combinatorial Principles ◽  
The Relationship ◽  
Mahlo Cardinals ◽  
Forcing Axiom

It is well known that many statements provable from combinatorial principles true in the constructible universe L can also be shown to be consistent with ZFC by forcing. Recent work by Shelah and Stanley [4] and the author [5] has clarified the relationship between the axiom of constructibility and forcing by providing Martin's Axiom-type forcing axioms equivalent to ◊ and the existence of morasses. In this paper we continue this line of research by providing a forcing axiom equivalent to □κ. The forcing axiom generalizes easily to inaccessible, non-Mahlo cardinals, and provides the motivation for a corresponding generalization of □κ.In order to state our forcing axiom, we will need to define a strategic closure condition for partial orders. Suppose P = 〈P, ≤〉 is a partial order. For each ordinal α we will consider a game played by two players, Good and Bad. The players choose, in order, the terms in a descending sequence of conditions 〈pβ∣β < α〉 Good chooses all terms pβ for limit β, and Bad chooses all the others. Bad wins if for some limit β<α, Good is unable to move at stage β because 〈pγ∣γ < β〉 has no lower bound. Otherwise, Good wins. Of course, we will be rooting for Good.

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