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

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 $\mathcal{L}_{\kappa,\kappa}$. Using this equivalence we show that if any $\kappa$-complete filter on $\lambda$ can be extended to a $\kappa$-complete ultrafilter and $\lambda^{<\kappa} = \lambda$ then $\square(\mu)$ fails for all regular $\mu\in[\kappa,2^\lambda]$. As an application, we improve the lower bound for the consistency strength of $\kappa$-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 $c.c.c.$ forcing adds a real and that for every uncountable regular cardinal $\kappa$, every $\kappa$-closed forcing of size $2^{<\kappa}$ collapses some cardinals.Comment: The proof of Lemma 6.3 has changed, and the large cardinal assumption used in earlier version is reduce

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

On the consistency of local and global versions of Chang's Conjecture

We show that for many pairs of infinite cardinals $\kappa > \mu^+ > \mu$, $(\kappa^{+}, \kappa)\twoheadrightarrow (\mu^+, \mu)$ is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal that $(\kappa^{+}, \kappa)\twoheadrightarrow (\mu^+, \mu)$ for every successor cardinal $\kappa$ and every $\mu < \kappa$, 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 $\omega$ many supercompact cardinals, that the super tree property holds at $\aleph_n$ for all $2 \leq n < \omega$ but there are weak square and a very good scale at $\aleph_{\omega}$

Destructibility of the tree property at ℵω+1\aleph_{\omega+1}

We construct a model in which the tree property holds in $\aleph_{\omega + 1}$ and it is destructible under $\text{Col}(\omega, \omega_1)$. 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 $\aleph_0$-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 $\aleph_{\omega^2+1}$ satisfies simultaneously a strong principle of reflection, called $\Delta$-reflection, and a version of the square principle, denoted $\square(\aleph_{\omega^2+1}).$ Thus we show that $\aleph_{\omega^2+1}$ can satisfy simultaneously a strong reflection principle and an anti-reflection principle