### A categorical characterization of sets among classes

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

The completion of a Suslin tree is shown to be a consistent example of a Corson compact L-space when endowed with the coarse wedge topology. The example has the further properties of being zero-dimensional and monotonically normal.

A dichotomy concerning ideals of countable subsets of some set is introduced and proved compatible with the Continuum Hypothesis. The dichotomy has influence not only on the Suslin Hypothesis or the structure of Hausdorff gaps in the quotient algebra $P\left(\mathbb{N}\right)$/ but also on some higher order statements like for example the existence of Jensen square sequences.

We define combinatorial structures which we refer to as flat morasses, and use them to construct a Lindelöf space with points ${G}_{\delta}$ of cardinality ${\aleph}_{\omega}$, consistent with GCH. The construction reveals, it is hoped, that flat morasses are a tool worth adding to the kit of any user of set theory.

We define two cardinal invariants of the continuum which arise naturally from combinatorially and topologically appealing properties of almost disjoint families of sets of the natural numbers. These are the never soft and never countably paracompact numbers. We show that these cardinals must both be equal to ${\omega}_{1}$ under the effective weak diamond principle $\u2666(\omega ,\omega ,<)$, answering questions of da Silva S.G., On the presence of countable paracompactness, normality and property $\left(a\right)$ in spaces from almost disjoint families,...

We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of ${C}^{s}\left(\kappa \right)$ and ${F}^{s}\left(\kappa \right)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂))...

Dopo una breve presentazione della teoria $A$, una teoria non riduzionista ed autoreferenziale dei fondamenti della matematica proposta da Clavelli, De Giorgi, Forti e Tortorelli nel 1987, si mostra l'inconsistenza di estensioni della teoria $A$ ottenute aggiungendo forti assiomi su relazioni e operazioni (ad es. assiomi che danno la composizione di operazioni, la congiunzione di relazioni, ecc.) e/o assiomi che forniscono qualche relazione "combinatoria".

We prove that-consistently-in the space ω* there are no P-sets with the ℂ-cc and any two fat P-sets with the ℂ⁺-cc are coabsolute.

Some relatives of the Juhász Club Principle are introduced and studied in the presence of CH. In particular, it is shown that a slight strengthening of this principle implies the existence of a Suslin tree in the presence of CH.

We prove that the statement: "there is a Corson compact space with a non-separable Radon measure" is equivalent to a number of natural statements in set theory.