Small filter forcing

1986 ◽  
Vol 51 (3) ◽  
pp. 526-546
Author(s):  
R. Michael Canjar
Keyword(s):  
Regular Cardinal ◽  
Ground Model ◽  
Random Real ◽  
Generic Subset ◽  
Partially Ordered ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Cohen Real ◽  
Basic Facts ◽  
Random Reals

Qλ is the set of nonprincipal filters on ω which are generated by fewer than λ sets, for λ a fixed uncountable, regular cardinal ≤ c. We analyze forcing with Qλ, where Qλ is partially ordered in such a way that a filter F1 is more informative than F2 iff F1 includes F2. Qλ-forcing adjoins an ultrafilter on ω but adds no new reals. We analyze Qλ-forcing from a forcing-theoretic viewpoint. We also analyze the properties of Qλ-generic ultrafilters. These properties are independent of ZFC and depend very much on the ground model. In particular, we study Qλ-forcing over ground models which are Cohen real extensions, random real extensions, and models which satisfy Martin's Axiom.In §2 we give notations and definitions, and review some of the basic facts about forcing and ultrafilters which we will use. In §3 we introduce Qλ-forcing and prove some basic lemmas about it. §4 studies Qc-forcing. §§5, 6, and 7 analyze Qλ-forcing over ground models of Martin's Axiom, ground models which are generated by Cohen reals, and ground models which are generated by random reals, respectively. Qλ-forcing over Cohen real and random real models is isomorphic to the notion of forcing which adjoins a Cohen generic subset of λ; this is proved in §8.

On a result of Szemerédi

2008 ◽  
Vol 73 (3) ◽  
pp. 953-956 ◽  
Author(s):  
Albin L. Jones
Keyword(s):  
Regular Cardinal ◽  
Short Proof ◽  
Unpublished Result ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Image Position

AbstractWe provide a short proof that if κ is a regular cardinal with κ < c, thenfor any ordinal α < min{, κ}. In particular,for any ordinal α < . This generalizes an unpublished result of E. Szemerédi that Martin's axiom implies thatfor any cardinal κ with κ < c.

The consistency of the axiom of universality for the ordering of cardinalities

1985 ◽  
Vol 50 (2) ◽  
pp. 502-509
Author(s):  
Marco Forti ◽  
Furio Honsell
Keyword(s):  
Set Theory ◽  
Partially Ordered Set ◽  
Regular Cardinal ◽  
Ordered Set ◽  
Partial Ordering ◽  
Generic Extension ◽  
Ground Model ◽  
Partially Ordered ◽  
Transfer Method ◽  
Image Position

T. Jech [4] and M. Takahashi [7] proved that given any partial ordering R in a model of ZFC there is a symmetric submodel of a generic extension of where R is isomorphic to the injective ordering on a set of cardinals.The authors raised the question whether the injective ordering of cardinals can be universal, i.e. whether the following axiom of “cardinal universality” is consistent:CU. For any partially ordered set (X, ≼) there is a bijection f:X → Y such that(i.e. x ≼ y iff ∃g: f(x) → f(y) injective). (See [1].)The consistency of CU relative to ZF0 (Zermelo-Fraenkel set theory without foundation) is proved in [2], but the transfer method of Jech-Sochor-Pincus cannot be applied to obtain consistency with full ZF (including foundation), since CU apparently is not boundable.In this paper the authors define a model of ZF + CU as a symmetric submodel of a generic extension obtained by forcing “à la Easton” with a class of conditions which add κ generic subsets to any regular cardinal κ of a ground model satisfying ZF + V = L.

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).

Sign in / Sign up

Export Citation Format

Share Document