Abstract
We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCℵ 0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℵ 1 is equivalent to the existence of a Kurepa tree of height ℵ 1 . Using the connection of BCκ 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 BCℵ 1 .
2. If it is consistent that BCℵ 1 , 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 ¬BCℵ ω +(∀n<ω)BCℵ n is consistent.
4. If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℵ ω .
5. If it is consistent that there is a 3-huge cardinal, then it is consistent thatBCκ for a proper class of cardinals κ of countable cofinality.
Original language | American English |
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Pages (from-to) | 168-184 |
Number of pages | 17 |
Journal | Journal of Symbolic Logic |
Volume | 78 |
Issue number | 1 |
DOIs | |
State | Published - Mar 2013 |
Keywords
- Borel Conjecture
- Chang's Conjecture
- Kurepa Hypothesis
- Rothberger bounded
- n-huge cardinal
EGS Disciplines
- Mathematics