On Sacks forcing and the Sacks property

2007 ◽  
pp. 95-139 ◽  
Author(s):  
Stefan Geschke ◽  
Sandra Quickert
Keyword(s):  
Sacks Forcing

Sacks Forcing

2017 ◽  
pp. 485-495
Author(s):  
Lorenz J. Halbeisen
Keyword(s):  
Sacks Forcing

scholarly journals Projective absoluteness for Sacks forcing

2009 ◽  
Vol 48 (7) ◽  
pp. 679-690 ◽  
Author(s):  
Daisuke Ikegami
Keyword(s):  
Sacks Forcing

scholarly journals Sacks forcing and the total failure of Martin's Axiom

1985 ◽  
Vol 19 (3) ◽  
pp. 211-225 ◽  
Author(s):  
James E. Baumgartner
Keyword(s):  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Sacks Forcing ◽  
Total Failure

scholarly journals Sacks forcing does not always produce a minimal upper bound

1979 ◽  
Vol 31 (1) ◽  
pp. 110-130 ◽  
Author(s):  
Fred G. Abramson
Keyword(s):  
Upper Bound ◽  
Sacks Forcing

sets and singletons

1999 ◽  
Vol 64 (2) ◽  
pp. 590-616
Author(s):  
Kai Hauser ◽  
W. Hugh Woodin
Keyword(s):  
Core Model ◽  
Real Numbers ◽  
The Third ◽  
Sacks Forcing ◽  
Projective Hierarchy ◽  
Definable Elements

AbstractWe extend work of H. Friedman, L. Harrington and P. Welch to the third level of the projective hierarchy. Our main theorems say that (under appropriate background assumptions) the possibility to select definable elements of non-empty sets of reals at the third level of the projective hierarchy is equivalent to the disjunction of determinacy of games at the second level of the projective hierarchy and the existence of a core model (corresponding to this fragment of determinacy) which must then contain all real numbers. The proofs use Sacks forcing with perfect trees and core model techniques.

The number of normal measures

2009 ◽  
Vol 74 (3) ◽  
pp. 1069-1080 ◽  
Author(s):  
Sy-David Friedmanc ◽  
Menachem Magidor
Keyword(s):  
Tuning Fork ◽  
Measurable Cardinal ◽  
Large Cardinal ◽  
Sacks Forcing ◽  
Treat All

AbstractThere have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH. where α is a cardinal at most κ++. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = α++, the maximum possible) and [1] (for α = κ+, after collapsing κ++). In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of Mitchell order α), [2] (as in [12], but where κ is the least measurable cardinal and α is less than κ, starting with a measurable of high Mitchell order) and [11] (as in [12], but where κ is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing. The proof uses κ-Sacks forcing and the “tuning fork” technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails.

scholarly journals Sacks forcing, Laver forcing, and Martin's axiom

1992 ◽  
Vol 31 (3) ◽  
pp. 145-161 ◽  
Author(s):  
Haim Judah ◽  
Arnold W. Miller ◽  
Saharon Shelah
Keyword(s):  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Sacks Forcing

Cardinal invariants and the collapse of the continuum by Sacks forcing

2008 ◽  
Vol 73 (2) ◽  
pp. 711-728
Author(s):  
Miroslav Repický
Keyword(s):  
Dominance Relation ◽  
Cardinal Invariant ◽  
Cardinal Invariants ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Sacks Forcing ◽  
Increasing Functions ◽  
Eventual Dominance ◽  
The Continuum ◽  
Families Of Subsets

AbstractWe study cardinal invariants of systems of meager hereditary families of subsets of ω connected with the collapse of the continuum by Sacks forcing and we obtain a cardinal invariant such that collapses the continuum to and . Applying the Baumgartner-Dordal theorem on preservation of eventually narrow sequences we obtain the consistency of . We define two relations and on the set (ωω)Fin of finite-to-one functions which are Tukey equivalent to the eventual dominance relation of functions such that if -unbounded, well-ordered by , and not -dominating, then there is a nonmeager p-ideal. The existence of such a system follows from Martin's axiom. This is an analogue of the results of [3], [9, 10] for increasing functions.

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