3 research outputs found
Large cardinals need not be large in HOD
We prove that large cardinals need not generally exhibit their large cardinal
nature in HOD. For example, a supercompact cardinal need not be weakly
compact in HOD, and there can be a proper class of supercompact cardinals in
, none of them weakly compact in HOD, with no supercompact cardinals in HOD.
Similar results hold for many other types of large cardinals, such as
measurable and strong cardinals.Comment: 20 pages. Commentary concerning this article can be made at
http://jdh.hamkins.org/large-cardinals-need-not-be-large-in-ho
Strongly compact cardinals and ordinal definability
This paper explores several topics related to Woodin's HOD conjecture. We
improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an
extendible cardinal to a strongly compact cardinal. We show that assuming there
is a strongly compact cardinal and the HOD hypothesis holds, there is no
elementary embedding from HOD to HOD, settling a question of Woodin. We show
that the HOD hypothesis is equivalent to a uniqueness property of elementary
embeddings of levels of the cumulative hierarchy. We prove that the HOD
hypothesis holds if and only if every regular cardinal above the first strongly
compact cardinal carries an ordinal definable omega-Jonsson algebra. We show
that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then
every supercompact cardinal is supercompact in HOD.Comment: 16 page
Structural Properties of the Stable Core
The stable core, an inner model of the form for
a simply definable predicate , was introduced by the first author in
[Fri12], where he showed that is a class forcing extension of its stable
core. We study the structural properties of the stable core and its
interactions with large cardinals. We show that the can
fail at all regular cardinals in the stable core, that the stable core can have
a discrete proper class of measurable cardinals, but that measurable cardinals
need not be downward absolute to the stable core. Moreover, we show that, if
large cardinals exist in , then the stable core has inner models with a
proper class of measurable limits of measurables, with a proper class of
measurable limits of measurable limits of measurables, and so forth. We show
this by providing a characterization of natural inner models for specially nested class clubs , like those arising in
the stable core, generalizing recent results of Welch [Wel19]