Related papers: On Shelah's Approachability Ideal
For infinite cardinals $\kappa,\lambda$ let $C(\kappa,\lambda)$ denote the class of all compact Hausdorff spaces of weight $\kappa$ and size $\lambda$. So $C(\kappa,\lambda)=\emptyset$ if $\kappa>\lambda$ or $\lambda>2^\kappa$. If F is a…
In [8] the second and third authors showed that if the least inaccessible cardinal is the least measurable cardinal, then there is an inner model with $o(\kappa)\geq2$. In this paper we improve this to $o(\kappa)\geq\kappa+1$ and show that…
We present two different types of models where, for certain singular cardinals lambda of uncountable cofinality, lambda -> (lambda, omega+1)^2, although lambda is not a strong limit cardinal. We announce, here, and will present in a…
Kunen's proof of the non-existence of Reinhardt cardinals opened up the research on very large cardinals, i.e., hypotheses at the limit of inconsistency. One of these large cardinals, I0, proved to have descriptive-set-theoretical…
Shelah-Woodin investigate the possibility of violating instances of $GCH$ through the addition of a single real. In particular they show that it is possible to obtain a failure of $CH$ by adding a single real to a model of $GCH$, preserving…
We succeed to say something on the identities of (mu^+, mu) when mu>theta>cf(mu), mu strong limit theta--compact. This hopefully will help to prove the consistency of ``some pair (mu^+,mu) is not compact'', however, this has not been…
We show that in the theory ZF + DC + for every cardinal {\lambda}, the set of infinite subsets of {\lambda} is well-ordered (i.e., Shelah's AX4), the {\theta}-function measuring the surjective size of the powersets P({\kappa}) can take…
We show that in Zermelo-Fraenkel Set Theory without the Axiom of Choice a surjectively modified continuum function $\theta(\kappa)$ can take almost arbitrary values for all infinite cardinals. This choiceless version of Easton's Theorem is…
We show that the existence of a well-known type of ideals on a regular cardinal $\lambda$ implies a compactness property concerning the specialisability of a tree of height $\lambda$ with no cofinal branches. We also use Neeman's method of…
Given a map $\phi:X\rightarrow Y$ between $F$-analytic manifolds over a local field $F$ of characteristic $0$, we introduce an invariant $\epsilon_{\star}(\phi)$ which quantifies the integrability of pushforwards of smooth compactly…
It is well-known that the automorphism towers of infinite centreless groups of cardinality kappa terminate in less than (2^{kappa})^+ steps. But an easy counting argument shows that (2^{kappa})^+ is not the best possible bound. However, in…
Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inaccessible in HOD.
We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V models ZFC + GCH is a given model (which in interesting cases contains instances of…
A ccc-generically supercompact cardinal $\kappa$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $\kappa$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically…
We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not…
Club guessing principles were introduced by Shelah as a weakening of Jensen's diamond. Most spectacularly, they were used to prove Shelah's ZFC bound on the power of the first singular cardinal. These principles have found many other…
For a ring R, denote by Spec^R_kappa(Gamma) the kappa-spectrum of the Gamma-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that Spec^R_{aleph_1}(Gamma) is full for suitable von…
Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T,kappa) be the number of isomorphism types of models of T of cardinality \kappa. We denote by \mu (respectively \hat\mu) the number of…
The~\emph{Rothberger number} $\mathfrak{b} (\mathcal{I})$ of a definable ideal $\mathcal{I}$ on $\omega$ is the least cardinal $\kappa$ such that there exists a Rothberger gap of type $(\omega,\kappa)$ in the quotient algebra $\mathcal{P}…
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…