Related papers: The Virtual Large Cardinal Hierarchy
We prove new instances of Halin's end degree conjecture (HC) in ZFC. In particular, we show that there is a proper class of cardinals kappa for which Halin's conjecture holds, answering two questions posed by Geschke, Kurkofka, Melcher, and…
The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The problem is closely related to the more…
An inaccessible cardinal kappa is supercompact when (kappa, lambda)-ITP holds for all lambda greater than or equal to kappa. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC…
We discuss some well-known compactness principles for uncountable structures of small regular sizes ($\omega_n$ for $2 \le n<\omega$, $\aleph_{\omega+1}$, $\aleph_{\omega^2+1}$, etc.), consistent from weakly compact (the size-restricted…
We show that from a supercompact cardinal \kappa, there is a forcing extension V[G] that has a symmetric inner model N in which ZF + not AC holds, \kappa\ and \kappa^+ are both singular, and the continuum function at \kappa\ can be…
The theory ZFC implies the scheme that for every cardinal $\delta$ we can make $\delta$ many dependent choices over any definable relation without terminal nodes. Friedman, the first author, and Kanovei constructed a model of ZFC$^-$ (ZFC…
We generalise the $\alpha$-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals $\alpha$ to arbitrary ordinals $\alpha$, and answer several questions posed in that paper. In particular, we show that $\alpha$-Ramseys are…
Let $\kappa$,$\lambda$ be regular uncountable cardinals such that $\lambda > \kappa^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(\kappa) = \lambda$ starting from a ground model in…
It is well-known that if $Y$ is a Banach space the $weak^{*}$-fragmentability of its dual space by some metric $\rho$ implies that $Y^{*}$ belongs to the Stegall class -- the former for shortly $\mathcal{W^*}F$, being the latter…
Through careful analysis of an argument of Brooke-Taylor and Rosicky, we show that the powerful image of any accessible functor is closed under colimits of $\kappa$-chains, $\kappa$ a sufficiently large almost measurable cardinal. This…
Let $\kappa,\lambda$ be regular cardinals, $\lambda\le\kappa$, let $\varphi$ be a sentence of the language $\mathcal L_{\kappa,\lambda}$ in a given signature, and let $\vartheta(\varphi)$ express the fact that $\varphi$ holds in a submodel,…
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…
We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal.…
It is known that the assumption that ``GCH first fails at \aleph_{\omega}'' leads to large cardinals in ZFC. Gitik and Koepke have demonstrated that this is not so in ZF: namely there is a generic cardinal-preserving extension of L (or any…
We give a brief survey on the interplay between forcing axioms and various other non-constructive principles widely used in many fields of abstract mathematics, such as the axiom of choice and Baire's category theorem. First of all we…
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the…
Building on work of Holy, L\"ucke and Njegomir \cite{MR3913154} on small embedding characterizations of large cardinals, we use some classical results of Baumgartner (see \cite{MR0384553} and \cite{MR0540770}), to give characterizations of…
We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections).…
We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem: Suppose $\kappa$ is a singular strong limit cardinal and…
We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma_1$-definability at uncountable regular cardinals. In particular we give its exact consistency strength firstly in terms of the second…