The linear refinement number and selection theory

• Autorzy: Michał Machura , Saharon Shelah , Boaz Tsaban
• Języki publikacji: EN

Abstrakty

EN

The linear refinement number 𝔩𝔯 is the minimal cardinality of a centered family in $[ω]^{ω}$ such that no linearly ordered set in $([ω]^{ω},⊆ *)$ refines this family. The linear excluded middle number 𝔩𝔵 is a variation of 𝔩𝔯. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that 𝔩𝔯 = 𝔩𝔵 = 𝔡 in all models where the continuum is at most ℵ₂, and that the cofinality of 𝔩𝔯 is uncountable. Using the method of forcing, we show that 𝔩𝔯 and 𝔩𝔵 are not provably equal to 𝔡, and rule out several potential bounds on these numbers. Our results solve a number of open problems.

Kategorie tematyczne

• 03E17: Cardinal characteristics of the continuum
• 03E75: Applications of set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2016
• Tom: 234
• Numer: 1
• Strony: 15-40

Daty

wydano

2016

Twórcy

autor

Michał Machura

• Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland
• Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel

autor

Saharon Shelah

• Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 9190401 Jerusalem, Israel
• Mathematics Department, Rutgers University, New Brunswick, NJ, U.S.A.

autor

Boaz Tsaban

• Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel
• Department of Mathematics, Weizmann Institute of Science, Rehovot 7610001, Israel

Identyfikatory

DOI

10.4064/fm124-8-2015