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Abstract
We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal κ of uncountable cofinality, while κ^+ enjoys various combinatorial properties.
As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal κ of uncountable cofinality where SCH fails and such that there is a collection of size less than 2^{κ^+} of graphs on κ^+ such that any graph on κ^+ embeds into one of the graphs in the collection.
As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal κ of uncountable cofinality where SCH fails and such that there is a collection of size less than 2^{κ^+} of graphs on κ^+ such that any graph on κ^+ embeds into one of the graphs in the collection.
Original language | English |
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Pages (from-to) | 7405-7441 |
Number of pages | 37 |
Journal | Transactions of the American Mathematical Society |
Volume | 369 |
Issue number | 10 |
Early online date | 31 May 2017 |
DOIs | |
Publication status | Published - Oct 2017 |
Keywords
- successor of singular cardinal
- iterated forcing
- strong chain condition
- Radin forcing forcing axiom
- universal graph
- indestructible supercompat cardinal
Profiles
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Mirna Dzamonja
- School of Engineering, Mathematics and Physics - Visiting Academic
Person: Other related - academic
Projects
- 1 Finished