Ordinal definable subsets of singular cardinals

James Cummings; Sy David Friedman; Menachem Magidor; Assaf Rinot; Dima Sinapova

doi:10.1007/s11856-018-1712-2

Ordinal definable subsets of singular cardinals

James Cummings, Sy David Friedman, Menachem Magidor, Assaf Rinot, Dima Sinapova

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

7 Scopus citations

Abstract

A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HOD_x contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal κ of countable cofinality κ⁺ is supercompact in HOD_x for all x ⊆ κ.

Original language	English
Pages (from-to)	781-804
Number of pages	24
Journal	Israel Journal of Mathematics
Volume	226
Issue number	2
DOIs	https://doi.org/10.1007/s11856-018-1712-2
State	Published - 1 Jun 2018
Externally published	Yes

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/s11856-018-1712-2

Cite this

@article{2916c868990f42da883f98d2140159cd,

title = "Ordinal definable subsets of singular cardinals",

abstract = "A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HODx contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal κ of countable cofinality κ+ is supercompact in HODx for all x ⊆ κ.",

author = "James Cummings and Friedman, {Sy David} and Menachem Magidor and Assaf Rinot and Dima Sinapova",

note = "Publisher Copyright: {\textcopyright} 2018, Hebrew University of Jerusalem.",

year = "2018",

month = jun,

day = "1",

doi = "10.1007/s11856-018-1712-2",

language = "English",

volume = "226",

pages = "781--804",

journal = "Israel Journal of Mathematics",

issn = "0021-2172",

publisher = "Springer New York",

number = "2",

}

TY - JOUR

T1 - Ordinal definable subsets of singular cardinals

AU - Cummings, James

AU - Friedman, Sy David

AU - Magidor, Menachem

AU - Rinot, Assaf

AU - Sinapova, Dima

PY - 2018/6/1

Y1 - 2018/6/1

N2 - A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HODx contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal κ of countable cofinality κ+ is supercompact in HODx for all x ⊆ κ.

AB - A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HODx contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal κ of countable cofinality κ+ is supercompact in HODx for all x ⊆ κ.

UR - http://www.scopus.com/inward/record.url?scp=85048305068&partnerID=8YFLogxK

U2 - 10.1007/s11856-018-1712-2

DO - 10.1007/s11856-018-1712-2

M3 - Article

AN - SCOPUS:85048305068

SN - 0021-2172

VL - 226

SP - 781

EP - 804

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 2

ER -

Ordinal definable subsets of singular cardinals

Abstract

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this