Year: 2011
Type: preprint
Abstract: We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $κ$, let {\sf BC}$_κ$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_κ$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (2)If it is consistent that ${\sf BC}_{\aleph_1}$ holds, then it is consistent that there is an inaccessible cardinal. (3)If it is consistent that there is a 1-inaccessible cardinal with $ω$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_ω} \, +\, (\forall n
Source: arXiv (Cornell University)
Authors: Galvin, Fred, Scheepers, Marion
Institutions: University of Kansas, Boise State University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2952494000