128 research outputs found
HOD, V and the GCH
Starting from large cardinals we construct a model of in which the
fails everywhere, but such that holds in its . The result
answers a question of Sy Friedman. Also, relative to the existence of large
cardinals, we produce a model of such that fails everywhere in
its .Comment: arXiv admin note: text overlap with arXiv:1510.0293
Fra\"iss\'e limit via forcing
Given a Fra\"{i}ss\'{e} class and an infinite cardinal
we define a forcing notion which adds a structure of size
using elements of , 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 with and
$\alpha^\omega > \alpha.
On a theorem of Magidor
Assuming is a supercompact cardinal and is an inaccessible
cardinal above it, we present an idea due to Magidor, to find a generic
extension in which 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 forcing
notion. We also show that if holds and 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 . 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 such that after forcing
with it, 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 at singular
cardinals . In particular, we answer questions of Erd\"{o}s-Hajnal [1]
and Todorcevic [6], [7] by showing that does not imply
nor the existence of a family of size such that has size for every .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
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