Collapsing $$\omega _2$$ ω 2 with semi-proper forcing

2017 ◽  
Vol 57 (1-2) ◽  
pp. 185-194 ◽  
Author(s):  
Stevo Todorcevic
Keyword(s):  
Proper Forcing

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).

Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom

2016 ◽  
Vol 56 (1-2) ◽  
pp. 1-20 ◽  
Author(s):  
Joan Bagaria ◽  
Victoria Gitman ◽  
Ralf Schindler
Keyword(s):  
Proper Forcing Axiom ◽  
Vopěnka’S Principle ◽  
Proper Forcing ◽  
Remarkable Cardinals ◽  
Forcing Axiom

Totally proper forcing and the Moore–Mrówka problem

2003 ◽  
Vol 177 (2) ◽  
pp. 121-137 ◽  
Author(s):  
Todd Eisworth
Keyword(s):  
Proper Forcing

A fragment of Asperó–Mota’s finitely proper forcing axiom and entangled sets of reals

2020 ◽  
Vol 251 (1) ◽  
pp. 35-68
Author(s):  
Tadatoshi Miyamoto ◽  
Teruyuki Yorioka
Keyword(s):  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Forcing Axiom

scholarly journals Hypergraphs and proper forcing

2019 ◽  
Vol 19 (02) ◽  
pp. 1950007
Author(s):  
Jindřich Zapletal
Keyword(s):  
Polish Space ◽  
Countable Collection ◽  
The Family ◽  
Proper Forcing

Given a Polish space [Formula: see text] and a countable collection of analytic hypergraphs on [Formula: see text], I consider the [Formula: see text]-ideal generated by Borel anticliques for the hypergraphs in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies.

scholarly journals Proper forcing remastered

2012 ◽  
pp. 331-362 ◽  
Author(s):  
Boban Veličcković ◽  
Giorgio Venturi
Keyword(s):  
Proper Forcing

scholarly journals On the equivalence of certain consequences of the proper forcing axiom

1995 ◽  
Vol 60 (2) ◽  
pp. 431-443 ◽  
Author(s):  
Peter Nyikos ◽  
Leszek Piątkiewicz
Keyword(s):  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Forcing Axiom

AbstractWe prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on ω1 with ω1 generators, then there exists an uncountable X ⊆ ω1, such that either [X]ω ∩ I = ∅ or [X]ω ⊆ I.

BPFA and projective well-orderings of the reals

2011 ◽  
Vol 76 (4) ◽  
pp. 1126-1136 ◽  
Author(s):  
Andrés Eduardo Caicedo ◽  
Sy-David Friedman
Keyword(s):  
Coding Scheme ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Forcing Axiom

AbstractIf the bounded proper forcing axiom BPFA holds and ω1 = ω1L, then there is a lightface Σ31 well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of “David's trick.” We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface Σ41 for many “consistently locally certified” relations R on ℝ. This is accomplished through a use of David's trick and a coding through the Σ2 stable ordinals of L.

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