HOD, V and the GCH

Starting from large cardinals we construct a model of $ZFC$ in which the $GCH$ fails everywhere, but such that $GCH$ holds in its $HOD$. The result answers a question of Sy Friedman. Also, relative to the existence of large cardinals, we produce a model of $ZFC+GCH$ such that $GCH$ fails everywhere in its $HOD$.Comment: arXiv admin note: text overlap with arXiv:1510.0293

Fra\"iss\'e limit via forcing

Given a Fra\"{i}ss\'{e} class $\mathcal{K}$ and an infinite cardinal $\kappa,$ we define a forcing notion which adds a structure of size $\kappa$ using elements of $\mathcal{K}$, which extends the Fra\"{i}ss\'{e} construction in the case $\kappa=\omega.

An introdution to forcing

The aim of these lectures is to give a short introduction to forcing. We will avoid metamathematical issues as much as possible and similarly we will avoid performing the actual construction of forcing. We assume familiarity with basic predicate logic, the axioms of ZF C set theory and constructible sets. We will also make use of tools like the coding of Borel sets and the Shoenfield absoluteness result

Singular cofinality conjecture and a question of Gorelic

We give an affirmative answer to a question of Gorelic \cite{Gorelic}, by showing it is consistent, relative to the existence of large cardinals, that there is a proper class of cardinals $\alpha$ with $cf(\alpha)=\omega_1$ and $\alpha^\omega > \alpha.

On a theorem of Magidor

Assuming $\kappa$ is a supercompact cardinal and $\lambda$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $\kappa=\aleph_\omega$ and $\lambda=\aleph_{\omega+1}.

An Easton like theorem in the presence of Shelah Cardinals

We show that Shelah cardinals are preserved under the canonical $GCH$ forcing notion. We also show that if $GCH$ holds and $F:REG\rightarrow CARD$ is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies $\forall \kappa\in REG,~ 2^{\kappa}=F(\kappa)$. This gives a partial answer to a question asked by Cody [1] and independently by Honzik [5]. We also prove an indestructibility result for Shelah cardinals

More on almost Souslin Kurepa trees

It is consistent that there exists a Souslin tree $T$ such that after forcing with it, $T$ becomes an almost Souslin Kurepa tree. This answers a question of Zakrzewski

Woodin's surgery method

In this short paper we give an overview of Woodin's surgery method

(Weak) diamond can fail at the least inaccessible cardinal

Starting from suitable large cardinals, we force the failure of (weak) diamond at the least inaccessible cardinal. The result improves an unpublished theorem of Woodin and a recent result of Ben-Neria, Garti and Hayut.Comment: This is the preliminary version of the pape

The generalized Kurepa hypothesis at singular cardinals

We discuss the generalized Kurepa hypothesis $KH_{\lambda}$ at singular cardinals $\lambda$. In particular, we answer questions of Erd\"{o}s-Hajnal [1] and Todorcevic [6], [7] by showing that $GCH$ does not imply $KH_{\aleph_\omega}$ nor the existence of a family $\mathcal{F} \subseteq [\aleph_\omega]^{\aleph_0}$ of size $\aleph_{\omega+1}$ such that $\mathcal{F} \restriction X$ has size $\aleph_0$ for every $X \subseteq S, |X|=\aleph_0$.Comment: In personal communication, Stevo Todorcevic informed the author that Theorem 3.1 has been obtained by him before; for example can be found on page 231 of his book Walks on ordinals and their characteristics. However our proof is different from hi