Related papers: Magidor cardinals
We study the filter versions of square bracket partition relations, focusing on Jonssonicity and Magidority. We show that the singular cardinals in a Kleinberg sequence above some strong partition cardinal are not Magidor, but the limit of…
We introduce a model-theoretic characterization of Magidor cardinals, from which we infer that Magidor filters are beyond ZFC-inconsistency
A group $G$ is J\'onsson if $|H| < |G|$ whenever $H$ is a proper subgroup of $G$. Using an embedding theorem of Obraztsov it is shown that there exists a J\'onsson group $G$ of infinite cardinality $\kappa$ if and only if there exists a…
We prove that $\alpha_M(\lambda)$ can be successor of a supercompact cardinal, when $\lambda$ is a Magidor cardinal. From this result we obtain the consistency of $\alpha_M(\lambda)$ being a successor of a singular cardinal with uncountable…
Improving a result of Woodin, we identify some classes of individually consistent but mutually inconsistent generic large cardinal axioms.
We give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor's identity crisis theorem for the first strongly compact cardinal.
We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of…
We introduce and axiomatize the notion of a reflective cardinal, use it to give semantics to higher order set theory, and explore connections between the notion of reflective cardinals and large cardinal axioms.
Let "ex" be the cardinality of the smallest independent family of subsets of omega (independent means that all nontrivial Boolean combinations are infinite) which cannot be extended to a homogeneous independent family. "Homogeneous" means…
Given an inaccessible J\'onsson cardinal $\lambda$, a sequence of results due to Shelah from Cardinal Arithmetic and Sh413 tell us that $\lambda$ must be at least $\lambda\times\omega$-Mahlo. We may then ask ourselves whether we can improve…
If it is consistent that there is a measurable cardinal, then it is consistent that all points g-delta Rothberger spaces have "small" cardinality.
We study the possible number of normal measures on a measurable cardinal in settings where inner model techniques are unavailable. Instead, we exploit consequences of the Ultrapower Axiom to obtain our theorems. We show that the classical…
The purpose of this paper is to provide an introductory overview of the large cardinal hierarchy in set theory. By a large cardinal, we mean any cardinal $\kappa$ whose existence is strong enough of an assumption to prove the consistency of…
We prove that on many inaccessible there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal lambda which is lambda x omega-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there…
We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and…
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 prove that if there is an elementary embedding from the universe to itself, then there is a proper class of measurable successor cardinals.
We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and omega-Erdos cardinals. They are characterized by the existence of "0^sharp-like" embeddings; however, they relativize…
According to Kearnes and Oman (2013), an ordered set $P$ is \emph{J\'onsson} if it is infinite and the cardinality of every proper initial segment of $P$ is strictly less than the cardinaliy of $P$. We examine the structure of J\'onsson…
Assuming $\kappa$ is a supercompact cardinal and $\lambda$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $\kappa=\aleph_\omega$ and $\lambda=\aleph_{\omega+1}.$