Fallen cardinals

Menachem Kojman; Saharon Shelah

doi:10.1016/S0168-0072(01)00045-8

Fallen cardinals

Menachem Kojman, Saharon Shelah

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra CompP_μ(μ) contains a complete subalgebra which is isomorphic to the collapse algebra CompCol(ω₁,μ^א₀). Consequently, adding a generic filter to the quotient algebra P_μ(μ)=P(μ)/[μ]^<μ collapses μ^א₀ to א₁. Another corollary is that the Baire number of the space U(μ) of all uniform ultrafilters over μ is equal to ω₂. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory.

Original language	English
Pages (from-to)	117-129
Number of pages	13
Journal	Annals of Pure and Applied Logic
Volume	109
Issue number	1-2
DOIs	https://doi.org/10.1016/S0168-0072(01)00045-8
State	Published - 15 May 2001

Keywords

03G05
04A10
04A20
54A25
54D80
54F65
Baire number
Boolean algebra
Distributivity
Forcing
Infinite cardinals
Pcf theory
Uniform ultrafilters

ASJC Scopus subject areas

Logic

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10.1016/S0168-0072(01)00045-8

Cite this

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title = "Fallen cardinals",

abstract = "We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra CompPμ(μ) contains a complete subalgebra which is isomorphic to the collapse algebra CompCol(ω1,μא0). Consequently, adding a generic filter to the quotient algebra Pμ(μ)=P(μ)/[μ]<μ collapses μא0 to א1. Another corollary is that the Baire number of the space U(μ) of all uniform ultrafilters over μ is equal to ω2. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory.",

keywords = "03G05, 04A10, 04A20, 54A25, 54D80, 54F65, Baire number, Boolean algebra, Distributivity, Forcing, Infinite cardinals, Pcf theory, Uniform ultrafilters",

author = "Menachem Kojman and Saharon Shelah",

note = "Funding Information: ∗Corresponding author. E-mail addresses: [email protected] (M. Kojman), [email protected] (S. Shelah). 1Partially supported by the Israeli Foundation for Basic Science. This paper has number 720 in the Shelah{\textquoteright}s list of publications.",

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doi = "10.1016/S0168-0072(01)00045-8",

language = "English",

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T1 - Fallen cardinals

AU - Kojman, Menachem

AU - Shelah, Saharon

N1 - Funding Information: ∗Corresponding author. E-mail addresses: [email protected] (M. Kojman), [email protected] (S. Shelah). 1Partially supported by the Israeli Foundation for Basic Science. This paper has number 720 in the Shelah’s list of publications.

PY - 2001/5/15

Y1 - 2001/5/15

N2 - We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra CompPμ(μ) contains a complete subalgebra which is isomorphic to the collapse algebra CompCol(ω1,μא0). Consequently, adding a generic filter to the quotient algebra Pμ(μ)=P(μ)/[μ]<μ collapses μא0 to א1. Another corollary is that the Baire number of the space U(μ) of all uniform ultrafilters over μ is equal to ω2. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory.

AB - We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra CompPμ(μ) contains a complete subalgebra which is isomorphic to the collapse algebra CompCol(ω1,μא0). Consequently, adding a generic filter to the quotient algebra Pμ(μ)=P(μ)/[μ]<μ collapses μא0 to א1. Another corollary is that the Baire number of the space U(μ) of all uniform ultrafilters over μ is equal to ω2. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory.

KW - 03G05

KW - 04A10

KW - 04A20

KW - 54A25

KW - 54D80

KW - 54F65

KW - Baire number

KW - Boolean algebra

KW - Distributivity

KW - Forcing

KW - Infinite cardinals

KW - Pcf theory

KW - Uniform ultrafilters

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U2 - 10.1016/S0168-0072(01)00045-8

DO - 10.1016/S0168-0072(01)00045-8

M3 - Article

AN - SCOPUS:0346312302

SN - 0168-0072

VL - 109

SP - 117

EP - 129

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 1-2

ER -

Fallen cardinals

Abstract

Keywords

ASJC Scopus subject areas

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