Related papers: Cardinal characteristics at aleph omega
We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_\omega$ together with the reflection of all stationary subsets of $\aleph_{\omega+1}$. This shows that two classic results of Magidor (from 1977 and…
We prove that if $\mu>{\rm cf}(\mu)=\omega$ and $2^\mu=\mu^+$ then $\binom{\mu^+}{\mu}\nrightarrow\binom{\mu^+\ \aleph_2}{\mu\ \mu}$.
In [CMRM24], it was proved that it is relatively consistent that \emph{bounding number} $\mathfrak{b}$ is smaller than the uniformity of $\mathcal{MA}$, where $\mathcal{MA}$ denotes the ideal of the meager-additive sets of $2^{\omega}$. To…
We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all…
We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e. the cofinality of lambda^lambda is strictly bigger than cov(meagre_lambda), i.e. the minimal number of nowhere dense subsets of…
In this paper, we study the notion of a generically extendible cardinal, which is a generic version of an extendible cardinal. We prove that the generic extendibility of $\omega_1$ or $\omega_2$ has small consistency strength, but that of a…
We address the question regarding the structure of the Mitchell order on normal measures. We show that every well founded order can be realized as the Mitchell order on a measurable cardinal $\kappa$ from some large cardinal assumption.
This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence $\phi$, in the sense that $\phi$ characterizes $\kappa$, if…
The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. If alpha(G-e) > alpha(G), then e is an alpha-critical edge, and if mu(G-e) < mu(G), then e is a mu-critical edge, where mu(G)…
We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e., the cofinality of ^{lambda}lambda, is strictly bigger than cov_lambda(meagre), i.e. the minimal number of nowhere dense subsets of…
In a recent publication [Ferreira {\it et al.}, Phys. Rev. D89 (2014) 083011] we tested the consistency of current astrophysical tests of the stability of the fine-structure constant $\alpha$ and the proton-to-electron mass ratio…
Characteristic earlier results were of the form CON$(2^{\aleph_0} \to [\lambda]^2_{n, 2})$, with $2^{\aleph_0} $ an ex-large cardinal, in the best case the first weakly Mahlo cardinal. Characteristic new results are CON$((2^{\aleph_0} =…
In the original version of this paper, we assume a theory $T$ that the logic $\mathbb L_{\kappa, \aleph_{0}}$ is categorical in a cardinal $\lambda > \kappa$, and $\kappa$ is a measurable cardinal. There we prove that the class of model of…
It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a…
We prove the consistency of $\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{p}=\mathfrak{g}=\mathfrak{s}<\mathrm{add}(\mathcal{M})=\mathrm{cof}(\mathcal{M})<\mathfrak{a}=\mathrm{non}(\mathcal{N})=\mathfrak{c}$ with ZFC where…
We improve previous work on the consistency strength of mutually stationary sequences of sets concentrating on points with divergent cofinality building on previous work by Adolf, Cox and Welch. Specifically, we have greatly reduced our…
The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for…
Dealing with the cardinal invariants p and t of the continuum we prove that m=p=aleph_2 -> t = aleph_1. In other words if MA_{aleph_1} (or a weak version of this) then (of course aleph_2 <= p <= t and) p = aleph_2 -> p = t . This is based…
Given any $\lambda\leq\kappa$, we construct a symmetric extension in which there is a set $X$ such that $\aleph(X)=\lambda$ and $\aleph^*(X)=\kappa$. Consequently, we show that $\mathsf{ZF}+$"For all pairs of infinite cardinals…
In this paper, we obtain the consistency, relative to large cardinals, of the existence of dense ideals on every successor of a regular cardinal simultaneously. Using a consequent transfer principle, we show that in this model there is a…