Related papers: On the derived models of self iterable universes

We show that the following two theories are equiconsistent: (T) ZFC, CH and "There is a dense ideal on the first uncountable cardinal such that if j is the generic embedding associated with it then its restriction on ordinals is independent…

Hellsten \cite{MR2026390} proved that when $\kappa$ is $\Pi^1_n$-indescribable, the \emph{$n$-club} subsets of $\kappa$ provide a filter base for the $\Pi^1_n$-indescribability ideal, and hence can also be used to give a characterization of…

The structure of automorphism groups of $\kappa$-existentially closed groups are studied by Kaya-Kuzucuo\u{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^\kappa$ whenever…

Let $\kappa$ be an inaccessible cardinal, $\mathfrak{U}$ be a universal algebra, and $\sim$ be the equivalence relation on $\mathfrak{U}^{\kappa}$ of eventual equality. From mild assumptions on $\kappa$ we give general constructions of…

Suppose that kappa is a singular cardinal of cofinality omega and GCH holds. Assume that for every n<omega the set of alphas with o(alpha)>= alpha^{+n} is unbounded in kappa.Then there is a cardinal preserving extension satisfying…

Assuming the existence of a proper class of supercompact cardinals, we force that for every regular cardinal $\kappa$, there are $\kappa^+$-Aronszajn trees and all such trees are special.

In the first part of the manuscript, we establish several consistency results concerning Woodin's $\HOD$ hypothesis and large cardinals around the level of extendibility. First, we prove that the first extendible cardinal can be the first…

Laver, and Woodin independently, showed that models of ${\rm ZFC}$ are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ${\rm ZFC}$,…

For an infinite cardinal $\kappa$, let $ded\kappa$ denote the supremum of the number of Dedekind cuts in linear orders of size $\kappa$. It is known that $\kappa<ded\kappa\leq 2^{\kappa}$ for all $\kappa$ and that $ded\kappa<2^{\kappa}$ is…

It is well-known that the automorphism towers of infinite centreless groups of cardinality kappa terminate in less than (2^{kappa})^+ steps. But an easy counting argument shows that (2^{kappa})^+ is not the best possible bound. However, in…

The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing…

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to $\omega_1$-sequences of the selection principle and…

An automaton is called reachable if every state is reachable from the initial state. This notion has been generalized coalgebraically in two ways: first, via a universal property on pointed coalgebras, namely, that a reachable coalgebra has…

We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal $J$ extending the nonstationary ideal…

Recently (Phys. Lett. A302 (2002) 253, hep-th/0208210; hep-th/0403146) employing bounded infinite-dimensional representations of the rotation group we have argued that one can obtain the consistent monopole theory with generalized Dirac…

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $\kappa$ with the property that the collection of all initial segments of the wellordering is definable by a…

We show that under $\BMM$ and "there exists a Woodin cardinal$"$, the nonstationary ideal on $\omega_1$ can not be defined by a $\Sigma_1$ formula with parameter $A \subset \omega_1$. We show that the same conclusion holds under the…

This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality $\lcf(\aleph_0, \de)$ of $\aleph_0$ modulo $\de$,…

In this paper we use infinitary Turing machines with tapes of length $\kappa$ and which run for time $\kappa$ as presented, e.g., by Koepke \& Seyfferth, to generalise the notion of type two computability to $2^{\kappa}$, where $\kappa$ is…

For certain uncountable cardinals $\kappa$ we produce a group of cardinality $\kappa$ which is freely indecomposable, strongly $\kappa$-free, and whose abelianization is free abelian of rank $\kappa$. The construction takes place in…