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

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

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.

scholarly journals Construction of dual modules using Martin's Axiom

2008 ◽  
Vol 320 (6) ◽  
pp. 2388-2404
Author(s):  
Rüdiger Göbel ◽  
Sebastian Pokutta
Keyword(s):  
Martin's Axiom ◽  
Martin’S Axiom

scholarly journals Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

2016 ◽  
Vol 68 (1) ◽  
pp. 44-66 ◽  
Author(s):  
David J. Fernández Bretón
Keyword(s):  
Abelian Group ◽  
Partial Orders ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Given

AbstractWe answer two questions of Hindman, Steprāns, and Strauss; namely, we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover, we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases; we also construct (assuming Martin's Axiom for countable partial orders, i.e., , a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.

The equivalence of a generalized Martin's axiom to a combinatorial principle

1981 ◽  
Vol 46 (4) ◽  
pp. 817-821 ◽  
Author(s):  
William Weiss
Keyword(s):  
Combinatorial Principle ◽  
Martin's Axiom ◽  
Martin’S Axiom

AbstractA generalized version of Martin's axiom, called BACH, is shown to be equivalent to one of its combinatorial consequences, a generalization of P(c).

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