85 research outputs found
Magidor-Malitz Reflection
In this paper we investigate the consequences and consistency of the downward
L\"owenheim-Skolem theorem for extension of the first order logic by the
Magidor-Malitz quantifier. We derive some combinatorial results and improve the
known upper bound for the consistency of Chang's Conjecture at successor of
singular cardinals
Restrictions on Forcings That Change Cofinalities
In this paper we investigate some properties of forcing which can be
considered "nice" in the context of singularizing regular cardinals to have an
uncountable cofinality. We show that such forcing which changes cofinality of a
regular cardinal, cannot be too nice and must cause some "damage" to the
structure of cardinals and stationary sets. As a consequence there is no
analogue to the Prikry forcing, in terms of "nice" properties, when changing
cofinalities to be uncountable.Comment: 8 pages; post-refereeing versio
Partial strong compactness and squares
In this paper we analyze the connection between some properties of partially
strongly compact cardinals: the completion of filters of certain size and
instances of the compactness of . Using this
equivalence we show that if any -complete filter on can be
extended to a -complete ultrafilter and
then fails for all regular . As an
application, we improve the lower bound for the consistency strength of
-compactness, a case which was explicitly considered by Mitchell
On Foreman's maximality principle
In this paper we consider the Foreman's maximality principle, which says that
any non-trivial forcing notion either adds a new real or collapses some
cardinals. We prove the consistency of some of its consequences. We prove that
it is consistent that every forcing adds a real and that for every
uncountable regular cardinal , every -closed forcing of size
collapses some cardinals.Comment: The proof of Lemma 6.3 has changed, and the large cardinal assumption
used in earlier version is reduce
On the consistency of local and global versions of Chang's Conjecture
We show that for many pairs of infinite cardinals ,
is consistent relative to
the consistency of a supercompact cardinal. We also show that it is consistent,
relative to a huge cardinal that for every successor cardinal and every ,
answering a question of Foreman.Comment: Fixed a proof for Lemma 4
The strong tree property and weak square
We show that it is consistent, relative to many supercompact
cardinals, that the super tree property holds at for all but there are weak square and a very good scale at
Destructibility of the tree property at
We construct a model in which the tree property holds in and it is destructible under . On the other
hand we discuss some cases in which the tree property is indestructible under
small or closed forcings
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
The tree property on a countable segment of successors of singular cardinals
Starting from the existence of many supercompact cardinals, we construct a
model of ZFC in which the tree property holds at a countable segment of
successor of singular cardinals
Square and Delta reflection
Starting from infinitely many supercompact cardinals, we force a model of ZFC
where satisfies simultaneously a strong principle of
reflection, called -reflection, and a version of the square principle,
denoted Thus we show that
can satisfy simultaneously a strong reflection principle and an anti-reflection
principle
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