Related papers: On equivalence relations Sigma_1^1-definable over …
The Hanf number for a set $S$ of sentences in $L_{\omega_1,\omega}$ (or some other logic) is the least infinite cardinal $\kappa$ such that for all $\varphi\in S$, if $\varphi$ has models in all infinite cardinalities less than $\kappa$,…
For an arbitrary infinite cardinal $\kappa$, we define classes of coordinatewise $\kappa$-slender and tailwise $\kappa$-slender modules as well as related classes of $h\kappa$-modules and initiate a study of these classes.
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…
It was noticed by Harel in [Har86] that "one can define $\Sigma_1^1$-complete versions of the well-known Post Correspondence Problem". We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a…
Ordinary infinitary languages L_{lambda, kappa} satisfy the Interpolation Theorem only in the case lambda <= {aleph_1}, kappa = {aleph_0}, this include first order logic of course. There are also some pairs of such logics satifying…
We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma$-Prikry. We show that given…
If $\kappa$ is regular and $2^{<\kappa}\leq\kappa^+$, then the existence of a weakly presaturated ideal on $\kappa^+$ implies $\square^*_\kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on…
We determine the large cardinal consistency strength of the existence of a $\lambda$-supercompact cardinal $\kappa$ such that GCH fails at $\lambda$. Indeed, we show that the existence of a $\lambda$-supercompact cardinal $\kappa$ such that…
An equivalence structure is a set with a single binary relation, satisfying sentences stating that the relation is an equivalence relation. A computable structure A is said to be $\Delta^0_\alpha$ categorical if for any computable structure…
We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and…
On every set A there is a rigid binary relation i.e. such a relation R \subseteq A \times A that there is no homomorphism (A,R) \rightarrow (A,R) except the identity (Vop{\v{e}}nka et al. [1965]). We prove that for each infinite cardinal…
We provide analogues of the results from [FMR11, CMMR13] in the reference list (which correspond to the case $\kappa = \omega$) for arbitrary $\kappa$-Souslin quasi-orders on any Polish space, for $\kappa$ an infinite cardinal smaller than…
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 $\lambda=\aleph_{\omega+1}.$
We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection…
This paper defines a Mitchell rank for supercompact cardinals. If $\kappa$ is a $\theta$-supercompact cardinal then $o_{\theta-sc}(\kappa) = \sup \{ o_{\theta-sc}(\mu) + 1 \ | \ \mu \in m(\kappa)\}$, where $m(\kappa)$ is the collection of…
If cf(kappa) = kappa, kappa^+< cf(lambda) = \lambda, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing…
The paper gives several sufficient conditions on the paracompactness of box products with an arbitrary number of many factors and boxes of arbitrary size. The former include results on generalised metrisability and Sikorski spaces. Of…
We give a level-by-level analysis of the Weak Vop\v{e}nka Principle for definable classes of relational structures (WVP), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level.…
We investigate whether the ultrafilter number function $\kappa \mapsto \mathfrak{u}(\kappa)$ on the cardinals is monotone, that is, whether $\mathfrak{u}(\lambda) \le \mathfrak{u}(\kappa)$ holds for all cardinals $\lambda < \kappa$ or not.…
Suppose $\kappa$ is $\lambda$-supercompact witnessed by an elementary embedding $j:V\rightarrow M$ with critical point $\kappa$, and further suppose that $F$ is a function from the class of regular cardinals to the class of cardinals…