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Chain conditions in maximal models

100%
Larson P., Todorčević S.
Fundamenta Mathematicae
|
2001
|
tom 168
|
nr 1
77-104
EN
We present two $ℙ_{max}$ 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 𝒦₃ fails. Of particular interest is the still open question whether 𝒦₂ holds in this model.
2
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Automorphisms of $𝓟(λ)/ℐ_{κ}$

100%
Larson P., McKenney P.
Fundamenta Mathematicae
|
2016
|
tom 233
|
nr 3
271-291
EN
We study conditions on automorphisms of Boolean algebras of the form $𝓟(λ)/ℐ_{κ}$ (where λ is an uncountable cardinal and $ℐ_{κ}$ is the ideal of sets of cardinality less than κ ) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every automorphism of $𝓟(2^{κ})/ℐ_{κ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $MA_{ℵ₁}$ implies both that every automorphism of 𝓟(ℝ)/Fin is trivial on a cocountable set and that every automorphism of 𝓟(ℝ)/Ctble is trivial.
3
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Universally measurable sets in generic extensions

81%
Larson P., Neeman I., Shelah S.
Fundamenta Mathematicae
|
2010
|
tom 208
|
nr 2
173-192
EN
A subset of a topological space is said to be universally measurable if it is measured by the completion of each countably additive σ-finite Borel measure on the space, and universally null if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality ℵ₁, and thus that there exist at least $2^{ℵ₁}$ such sets. Laver showed in the 1970's that consistently there are just continuum many universally null sets of reals. The question of whether there exist more than continuum many universally measurable sets of reals was asked by Mauldin in 1978. We show that consistently there exist only continuum many universally measurable sets. This result also follows from work of Ciesielski and Pawlikowski on the iterated Sacks model. In the models we consider (forcing extensions by suitably-sized random algebras) every set of reals is universally measurable if and only if it and its complement are unions of ground model continuum many Borel sets.
4
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Guessing clubs in the generalized club filter

81%
König B., Larson P., Yoshinobu Y.
Fundamenta Mathematicae
|
2007
|
tom 195
|
nr 2
177-191
EN
We present principles for guessing clubs in the generalized club filter on $𝓟_{κ}λ$. These principles are shown to be weaker than classical diamond principles but often serve as sufficient substitutes. One application is a new construction of a λ⁺-Suslin-tree using assumptions different from previous constructions. The other application partly solves open problems regarding the cofinality of reflection points for stationary subsets of $[λ]^{ℵ₀}$.
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