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
N1 - Publisher Copyright:
© 2018, Hebrew University of Jerusalem.
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
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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 -