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 2κ+ 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) | 7405-7411 |
Number of pages | 7 |
Journal | Transactions of the American Mathematical Society |
Volume | 369 |
Issue number | 10 |
DOIs | |
State | Published - 2017 |
Bibliographical note
Publisher Copyright:© 2017 American Mathematical Society.
Keywords
- Forcing axiom
- Indestructible supercompact cardinal
- Iterated forcing
- Radin forcing
- Strong chain condition
- Successor of singular cardinal
- Universal graph