Related papers: Magidor cardinals
We extend a transitive model V of ZFC + GCH cardinal preservingly to a model N of ZF + "GCH holds below Alef_omega" + "there is a surjection from the power set of Alef_omega onto lambda" where lambda is an arbitrarily high fixed cardinal in…
Continuing \cite{GitJir22}, we develop a version of Extender-based Magidor-Radin forcing where there are no extenders on the top ordinal. As an application, we provide another approach to obtain a failure of SCH on a club subset of an…
This report presents an expression for the number of a multiset's sub-multisets of a given cardinality as a function of the multiplicity of its elements. This is also the number of distinct samples of a given size that may be produced by…
We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from…
Assume $\mathsf{ZF}+\mathsf{AD}+V=L(\mathbb{R})$ and let $\kappa<\Theta$ be an uncountable cardinal. We show that $\kappa$ is J\'onsson, and that if $\mathrm{cof}(\kappa)=\omega$ then $\kappa$ is Rowbottom. We also establish some other…
In this survey, we collect necessary conditions for the successor of a singular cardinal to be Jonsson.
Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuum problem and identify structural features of the levels…
Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a…
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 show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. In…
Assuming the existence of a strong cardinal, we find a model of ZFC in which for each uncountable regular cardinal $\lambda,$ there is no universal graph of size $\lambda$.
In [Bon20], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and…
Exacting and ultraexacting cardinals are large cardinal numbers compatible with the Zermelo-Fraenkel axioms of set theory, including the Axiom of Choice. In contrast with standard large cardinal notions, their existence implies that the…
It is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) < \mathrm{add}(\mathrm{Meager})= \mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) = 2^{\aleph_0}. \] Assuming four…
Despite being an established notion in the large cardinal hierarchy, results about Woodin cardinals are sparse in the literature. Here we gather known results about the preservation of Woodin cardinals under certain forcing extensions, as…
The aim of this paper is a characterization of great antipodal sets of complex Grassmannian manifolds as certain designs with the smallest cardinalities.
We study several cardinal characteristics of closed graphs G on compact metrizable spaces. In particular, we address the question when it is consistent for the bounding number to be strictly smaller than the smallest size of a set not…
Given an uncountable cardinal $\kappa$, we consider the question of whether subsets of the power set of $\kappa$ that are usually constructed with the help of the Axiom of Choice are definable by $\Sigma_1$-formulas that only use the…
Suppose there is a Reinhardt cardinal. Then (1) $M_n(X)$ exists and is fully iterable (above $X$) for every transitive set $X$ and every $n<\omega$ (here $M_n(X)$ denotes the canonical minimal proper class inner model containing $X$ and…
We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly kappa+ many normal measures on the least measurable cardinal kappa. This answers a question of Stewart Baldwin. The methods…