Non-constructive Galois-Tukey connections

Mildenberger, Heike

doi:10.2307/2275635

Cited by 8 publications

(11 citation statements)

References 3 publications

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“…However it might be difficult. In [11] and [12], Mildenberger introduced another variation of reaping numbers r n and r n = r m (= r) holds for n, m ∈ ω but it is proved that there are no nice Galois-Tukey connections between Mildenberger's reaping numbers. .…”

Section: N-splitting Number and N-reaping Numbermentioning

confidence: 99%

Pair-splitting, pair-reaping and cardinal invariants ofFσ-ideals

Hrušák

Meza-Alcántara

Minami³

2010

J. symb. log.

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Abstract. We investigate the pair-splitting number s pair which is a variation of splitting number, pair-reaping number r pair which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of F σ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

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Section: N-splitting Number and N-reaping Numbermentioning

confidence: 99%

Pair-splitting, pair-reaping and cardinal invariants ofFσ-ideals

Hrušák

Meza-Alcántara

Minami³

2010

J. symb. log.

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“…In [8], the first example of a correct inequality between the norms of two relations without a Borel morphism proving it was given. The older examples for non-existence of Borel morphisms are based upon incorrect inequalities and forcing, see [4] and Section 2 of this paper.…”

Section: Introductionmentioning

confidence: 99%

“…In [8] we have shown that there are pairs of relations with no Borel morphism connecting them. The reason was a strong impact of the first of the two functions that constitute a morphism, the so-called function on the questions.…”

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confidence: 99%

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confidence: 99%

“…(n" n (0 \ M ) ,[u]", ((0 \ M ) x [u]") n R i ) , cf [8],. so the premise in the defining implication is never fulfilled.2.…”

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confidence: 99%

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Borel on the Questions Versus Borel on the Answers

Mildenberger

1999

Mathematical Logic Qtrly

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We consider morphisms (also called Galois-Tukey connections) between binary relations that are used in the theory of cardinal characteristics. In [8] we have shown that there are pairs of relations with no Borel morphism connecting them. The reason was a strong impact of the first of the two functions that constitute a morphism, the so-called function on the questions. In this work we investigate whether the second half, the function on the answers' side, has a similarly strong impact. The main question is: Does the nonexistence of a Borel morphism imply the non-existence of a morphism that is only Borel on the answers' side? We give sufficient conditions for an affirmative answer. The results are applied to the unsplitting relations where it has been open whether there is a morphism that is Borel on the answers' side.Mathematics Subject Classification: 03E15, 03335, 03355.

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No Borel Connections for the Unsplitting Relations

Mildenberger

2002

Mathematical Logic Quarterly

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We prove that there is no Borel connection for non-trivial pairs of unsplitting relations. This was conjectured in [3].Mathematics Subject Classification: 03E15, 03E17, 03E35. Vojtáš [6] introduced a framework in which cardinal characteristics of the continuum can be regarded as norms of corresponding relationsA Galois-Tukey connection from a relation B to a relation A, which we call as in [1] a morphism from A to B, is a pair of functions (α, β) such thatIf there is a morphism from A to B, then B ≤ A , and indeed the proofs of the inequalities usually exhibit morphisms between the corresponding relations. We deal with the unsplitting relations: For n ≥ 1, we haver 2 is the usual unsplitting number. It is easy to see that r m = r n for m, n ≥ 2.In [3] it was proved that there is no morphism for the sharp unsplitting relations with Baire measurable first component, and it was conjectured that the same is true for the ordinary unsplitting relations. In this paper we prove this conjecture, under the additional premise that α and β are Lebesgue measurable.The present work is built partly upon techniques from the mentioned work on the sharp unsplitting relations, to which we add some additional steps mainly coming from 1) The author was supported by a Minerva fellowship. 2)

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Non-constructive Galois-Tukey connections

Abstract: There are inequalities between cardinal characteristics of the continuum that are true in any model of ZFC, but without a Borel morphism proving the inequality. We answer some questions from Blass [1].

Cited by 8 publications

References 3 publications

Pair-splitting, pair-reaping and cardinal invariants ofFσ-ideals

Pair-splitting, pair-reaping and cardinal invariants ofFσ-ideals

Borel on the Questions Versus Borel on the Answers

No Borel Connections for the Unsplitting Relations

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