Borel's Conjecture in Topological Groups
| Autor: | Marion Scheepers, Fred Galvin |
|---|---|
| Rok vydání: | 2011 |
| Předmět: |
$n$-huge cardinal
Logic Mathematics::General Topology Chang's Conjecture Group Theory (math.GR) Topological space Borel conjecture Separable space Null set Borel hierarchy Inaccessible cardinal FOS: Mathematics Borel measure Mathematics - General Topology Mathematics Discrete mathematics 03E05 03E35 03E55 03E65 22A99 Rothberger bounded General Topology (math.GN) Mathematics - Logic Primary 03E05 Secondary 03E35 03E55 03E65 22A99 Philosophy Mathematics::Logic Borel Conjecture Kurepa Hypothesis Borel set Logic (math.LO) Mathematics - Group Theory |
| Zdroj: | J. Symbolic Logic 78, iss. 1 (2013), 168-184 |
| DOI: | 10.48550/arxiv.1107.5383 |
| Popis: | We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let {\sf BC}$_{\kappa}$ 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 ${\sf BC}_{\kappa}$ 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 $\omega$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_{\omega}} \, +\, (\forall n Comment: 15 pages |
| Databáze: | OpenAIRE |
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