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A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

Miloš S. Kurilić, Aleksandar Pavlović (2014)

Czechoslovak Mathematical Journal

We compare the forcing-related properties of a complete Boolean algebra 𝔹 with the properties of the convergences λ s (the algebraic convergence) and λ ls on 𝔹 generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that λ ls is a topological convergence iff forcing by 𝔹 does not produce new reals and that λ ls is weakly topological if 𝔹 satisfies condition ( ) (implied by the 𝔱 -cc). On the other hand, if λ ls is a weakly topological convergence, then 𝔹 is a 2 𝔥 -cc algebra...

A note on a question of Abe

Douglas Burke (2000)

Fundamenta Mathematicae

Assuming large cardinals, we show that every κ-complete filter can be generically extended to a V-ultrafilter with well-founded ultrapower. We then apply this to answer a question of Abe.

Another ⋄-like principle

Michael Hrušák (2001)

Fundamenta Mathematicae

A new ⋄-like principle consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that ¬ is consistent with CH and that in many models of = ω₁ the principle holds. As implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁ in such models....

Balcar's theorem on supports

Lev Bukovský (2018)

Commentationes Mathematicae Universitatis Carolinae

In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6] B. Balcar showed that if σ D M is a support, M being an inner model of ZFC, and 𝒫 ( D σ ) M = r ` ` σ with r M , then r determines a preorder " " of D such that σ becomes a filter on ( D , ) generic over M . We show that if the relation r is replaced by a function 𝒫 ( D σ ) M = f - 1 ( σ ) , then there exists an equivalence relation " " on D and a partial order on D / such that D / is a complete Boolean algebra, σ / is a generic filter and [ f ( u ) ] = - ( u / ) for any u D , u M .

Chain conditions in maximal models

Paul Larson, Stevo Todorčević (2001)

Fundamenta Mathematicae

We present two m a x varations which create maximal models relative to certain counterexamples to Martin’s Axiom, in hope of separating certain classical statements which fall between MA and Suslin’s Hypothesis. One of these models is taken from [19], in which we maximize relative to the existence of a certain type of Suslin tree, and then force with that tree. In the resulting model, all Aronszajn trees are special and Knaster’s forcing axiom ₃ fails. Of particular interest is the still open question...

Cohen real and disjoint refinement of perfect sets

Miroslav Repický (2000)

Commentationes Mathematicae Universitatis Carolinae

We prove that if there exists a Cohen real over a model, then the family of perfect sets coded in the model has a disjoint refinement by perfect sets.

Coherent adequate sets and forcing square

John Krueger (2014)

Fundamenta Mathematicae

We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on ω₂ using finite conditions.

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