Related papers: A Note on Singular Cardinals in Set Theory Without…
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…
Assuming G\"odel's axiom of constructibility $V=L$, we construct a $\chi$-free abelian group $G$ of singular cardinality for some suitable cardinal $\chi$ which is regular and uncountable, equipped with the property that for every…
A new type of singularity theorem, based on spatial averages of physical quantities, is presented and discussed. Alternatively, the results inform us of when a spacetime can be singularity-free. This theorem provides a decisive…
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…
We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak…
Addressing a question of Erdos and Hajnal we show that one can force a graph with no large independent sets and no monochromatic triples as successors of strong limit singular cardinals, even without assuming GCH. The result can be pushed…
The relationship between the large cardinal notions of strong compactness and supercompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis called the Ultrapower Axiom, we prove that the notions are…
This paper introduces a novel binary stability property for voting rules-called binary self-selectivity-by which a society considering whether to replace its voting rule using itself in pairwise elections will choose not to do so. In…
In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu$ such that the singular cardinal hypothesis fails at $\nu$ and every collection of fewer than $\mathrm{cf}(\nu)$…
In set theory without the Axiom of Choice, we study the possible placement of Erdos-Dushnik-Miller theorem restricted to an uncountable set of vertices in the hierarchy of weak choice forms. We also answer a part of a question raised by…
We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally…
The necessity and benefit of singular solutions in the study of physical systems is shown. By singular solutions we mean solutions that are not contained in the general solution of the system of equations that describes the dynamic system…
The paper is a first of two and aims to show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic…
Many set theorists point to the linearity phenomenon in the hierarchy of consistency strength, by which natural theories tend to be linearly ordered and indeed well ordered by consistency strength. Why should it be linear? In this paper I…
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…
In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $\mathbf{ZF}$, some are shown to be independent of…
We prove, via transfinite recursion, the existence, inside any linearly ordered set of appropriate regular cardinality $\lambda$, of a particular kind of well-ordered subsets characterized by the property of $\lambda$-fullness. Let $H$ be a…
We show that the consistency of $\mathrm{ZF} + \mathrm{AD}_{\mathbb{R}} + ``\Theta$ is measurable$"$ implies the consistency of $\mathrm{ZF} +``\Theta$ is the least strongly regular cardinal and the least measurable cardinal$"$ + $``$all…
We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of…
For a strongly inacessible cardinal $\kappa$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}\kappa$-box product topology - the ideal of "null" sets in the sense of [Sh:1004]…