8,029 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
Heights of Models of and the Existence of End Elementary Extensions
The existence of End Elementary Extensions of models M of ZFC is related to
the ordinal height of M, according to classical results due to Keisler, Morley
and Silver. In this paper, we further investigate the connection between the
height of M and the existence of End Elementary Extensions of M. In particular,
we prove that the theory `ZFC + GCH + there exist measurable cardinals + all
inaccessible non weakly compact cardinals are possible heights of models with
no End Elementary Extensions' is consistent relative to the theory `ZFC + GCH +
there exist measurable cardinals + the weakly compact cardinals are cofinal in
ON'. We also provide a simpler coding that destroys GCH but otherwise yields
the same result
Destruction or Preservation As You Like It
The Gap Forcing Theorem, a key contribution of this paper, implies
essentially that after any reverse Easton iteration of closed forcing, such as
the Laver preparation, every supercompactness measure on a supercompact
cardinal extends a measure from the ground model. Thus, such forcing can create
no new supercompact cardinals, and, if the GCH holds, neither can it increase
the degree of supercompactness of any cardinal; in particular, it can create no
new measurable cardinals. In a crescendo of what I call exact preservation
theorems, I use this new technology to perform a kind of partial Laver
preparation, and thereby finely control the class of posets which preserve a
supercompact cardinal. Eventually, I prove the `As You Like It' Theorem, which
asserts that the class of -directed closed posets which preserve a
supercompact cardinal can be made by forcing to conform with any
pre-selected local definition which respects the equivalence of forcing. Along
the way I separate completely the levels of the superdestructibility hierarchy,
and, in an epilogue, prove that the notions of fragility and
superdestructibility are orthogonal---all four combinations are possible.Comment: This paper appeared in 1998; I am finally now uploading it to the
arxiv. 45 pages. Commentary can be made at
http://jdh.hamkins.org/asyoulikeit
Forcing in Ramsey theory
Ramsey theory and forcing have a symbiotic relationship. At the RIMS
Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave
three tutorials on Ramsey theory in forcing. The first two tutorials
concentrated on forcings which contain dense subsets forming topological Ramsey
spaces. These forcings motivated the development of new Ramsey theory, which
then was applied to the generic ultrafilters to obtain the precise structure
Rudin-Keisler and Tukey orders below such ultrafilters. The content of the
first two tutorials has appeared in an expository article submitted to the
SEALS 2016 Proceedings. The third tutorial concentrated on uses of forcing to
prove Ramsey theorems for trees which are applied to determine big Ramsey
degrees of homogeneous relational structures. This is the focus of this paper.Comment: 17 pages, Expository article. A few typos corrected. Journal of
Symbolic Logic, 202
Killing the GCH everywhere with a single real
Shelah-Woodin investigate the possibility of violating instances of
through the addition of a single real. In particular they show that it is
possible to obtain a failure of by adding a single real to a model of
, preserving cofinalities. In this article we strengthen their result by
showing that it is possible to violate at all infinite cardinals by
adding a single real to a model of Our assumption is the existence of an
-strong cardinal, by work of Gitik and Mitchell it is known
that more than an -strong cardinal is required
Joint diamonds and Laver diamonds
The concept of jointness for guessing principles, specifically
and various Laver diamonds, is introduced. A family of
guessing sequences is joint if the elements of any given sequence of targets
may be simultaneously guessed by the members of the family. While equivalent in
the case of , joint Laver diamonds are nontrivial new
objects. We give equiconsistency results for most of the large cardinals under
consideration and prove sharp separations between joint Laver diamonds of
different lengths in the case of -supercompact cardinals.Comment: 34 pages; revised version with several improvements, including
expanded Sections 3.3 and
The structure of the Mitchell order - II
We address the question regarding the structure of the Mitchell order on
normal measures. We show that every well founded order can be realized as the
Mitchell order on a measurable cardinal from some large cardinal
assumption
Magidor cardinals
We define Magidor cardinals as J\'onsson cardinals upon replacing colorings
of finite subsets by colorings of -bounded subsets. Unlike J\'onsson
cardinals which appear at some low level of large cardinals, we prove the
consistency of having quite large cardinals along with the fact that no Magidor
cardinal exists
Rigid ideals
An ideal on a cardinal is called \emph{rigid} if all
automorphisms of are trivial. An ideal is called
\emph{-minimal} if whenever is generic and , it follows that . We prove that the
existence of a rigid saturated -minimal ideal on , where is a
regular cardinal, is consistent relative to the existence of large cardinals.
The existence of such an ideal implies that GCH fails. However, we show that
the existence of a rigid saturated ideal on , where is an
\emph{uncountable} regular cardinal, is consistent with GCH relative to the
existence of an almost-huge cardinal. Addressing the case , we show
that the existence of a rigid \emph{presaturated} ideal on is
consistent with CH relative to the existence of an almost-huge cardinal. The
existence of a \emph{precipitous} rigid ideal on where is an
uncountable regular cardinal is equiconsistent with the existence of a
measurable cardinal
Descriptive inner model theory
A paper for general audience about descriptive inner model theory.Comment: To appear in BS
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