More on the Boolean Prime Ideal Theorem
We prove the consistency of Zermelo--Fraenkel set theory with the Axiom of Dependent Choices, no Vitali sets and a large fragment of the Boolean Prime Ideal Theorem.
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arXiv subjects
Jindrich Zapletal
Publications and source records attributed to Jindrich Zapletal.
Independence relations in the Solovay model II
I provide a novel axiomatization of the Solovay model and a purely geometric treatment of the theory of balanced forcing.
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Independence relations in the Solovay model I
I provide a novel geometric axiomatization of the Solovay model. This serves as a vehicle for concise and forcing-free proofs of classical results in the model, as well as a tool for a purely geometric development of the theory of balanced forcing.
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Nonarchimedean groups and the axiom of choice
We prove several novel connections between properties of nonarchimedean groups and fragments of the axiom of choice which hold in their associated permutation model.
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On the Isbell problem
We present three models concerning Tukey types of ultrafilters on $ω$. The first model is built via a countable support iteration, and we show there is no basically generated ultrafilter in such model. The second and third models are built upon different and novel techniques, and in such models all ultrafilters are Tukey top, thus providing an answer to the Isbell problem. In all models there is no $\mathsf{nwd}$-ultrafilter.
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Dynamical ideals and the axiom of choice
I provide several natural properties of group actions which translate into fragments of axiom of choice in the associated permutation models of choiceless set theory.
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Coloring equilateral triangles
It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles on a given Euclidean space has countable chromatic number, while the hypergraph of isosceles triangles in the plane does not.
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On the Consistency Strength of MM($ω_1$)
We prove that the consistency strength of Martin's Maximum restricted to partial orders of cardinality $ω_1$ follows from the consistency of ZFC.
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Triangles and Vitali sets
It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles in Euclidean plane has countable chromatic number, while there is no Vitali set.
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Coloring triangles and rectangles
It is consistent that ZF+DC holds, the hypergraph of rectangles on a given Euclidean space has countable chromatic number, while the hypergraph of equilateral triangles in two-dimensional Euclidean space does not.
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Noetherian spaces in choiceless set theory
I prove several independence results in the choiceless ZF+DC theory which separate algebraic and non-algebraic consequences of the axiom of choice.
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Coloring closed Noetherian graphs
If G is a closed Noetherian graph on a sigma-compact Polish space without an infinite clique, it is consistent with the choiceless set theory ZF+DC that G is countably chromatic and there is no Vitali set.
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Coloring redundant algebraic hypergraphs
We prove several consistency results in choiceless set theory ZF+DC regarding countable chromatic numbers of various algebraic hypergraphs on Euclidean spaces.
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Coloring the distance graphs
Let n>0 be a number. Let Gn be the graph on n-dimensional Euclidean space connecting points of rational distance. It is consistent with the choiceless theory ZF+DC that Gn has countable chromatic number yet Gn+1 does not.
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Krull dimension in set theory
Let n>1 be a number. Let Gn be the hypergraph of all rectangles in an n-dimensional Euclidean space. It is consistent that ZF+DC holds, the chromatic number of Gn is countable, yet the chromatic number of Gn+1 is uncountable.
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Transcendental pairs of generic extensions
We isolate a new preservation class of Suslin forcings and prove several associated consistency results in the choiceless theory ZF+DC regarding countable chromatic numbers of various Borel hypergraphs.
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Coloring the distance graph in three dimensions
Let Gn be the graph on n-dimensional Euclidean space connecting points of rational Euclidean distance. It is consistent relative to an inaccessible cardinal that ZF+DC holds and G3 has countable chromatic number, yet G4 has uncountable chromatic number.
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Sequential topologies and Dedekind finite sets
It is consistent with ZF set theory that the Euclidean topology on the real line is not sequential, yet every infinite set of reals contains a countably infinite subset. This answers a question of Gutierres.
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