A framework for forcing constructions at successors of singular cardinals

James Cummings; Mirna Džamonja; Menachem Magidor; Charles Morgan; Saharon Shelah

doi:10.1090/tran/6974

A framework for forcing constructions at successors of singular cardinals

James Cummings
, Mirna Džamonja
, Menachem Magidor
, Charles Morgan
, Saharon Shelah

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

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.

Original language	English
Pages (from-to)	7405-7411
Number of pages	7
Journal	Transactions of the American Mathematical Society
Volume	369
Issue number	10
DOIs	https://doi.org/10.1090/tran/6974
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

Access to Document

10.1090/tran/6974

Cite this

@article{02df86d00f5a4b1582e662a67e3cee04,

title = "A framework for forcing constructions at successors of singular cardinals",

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.",

keywords = "Forcing axiom, Indestructible supercompact cardinal, Iterated forcing, Radin forcing, Strong chain condition, Successor of singular cardinal, Universal graph",

author = "James Cummings and Mirna D{\v z}amonja and Menachem Magidor and Charles Morgan and Saharon Shelah",

note = "Publisher Copyright: {\textcopyright} 2017 American Mathematical Society.",

year = "2017",

doi = "10.1090/tran/6974",

language = "אנגלית",

volume = "369",

pages = "7405--7411",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "10",

}

TY - JOUR

T1 - A framework for forcing constructions at successors of singular cardinals

AU - Cummings, James

AU - Džamonja, Mirna

AU - Magidor, Menachem

AU - Morgan, Charles

AU - Shelah, Saharon

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Forcing axiom

KW - Indestructible supercompact cardinal

KW - Iterated forcing

KW - Radin forcing

KW - Strong chain condition

KW - Successor of singular cardinal

KW - Universal graph

UR - https://www.scopus.com/pages/publications/85027026751

U2 - 10.1090/tran/6974

DO - 10.1090/tran/6974

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85027026751

SN - 0002-9947

VL - 369

SP - 7405

EP - 7411

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 10

ER -

A framework for forcing constructions at successors of singular cardinals

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this