Related papers: Selective Independence and $h$-Perfect Tree Forcin…
The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a…
A perfect forest is a spanning forest of a connected graph $G$, all of whose components are induced subgraphs of $G$ and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a…
Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary…
Assuming $\rm PFA$, we shall use internally club $\omega_1$-guessing models as side conditions to show that for every tree $T$ of height $\omega_2$ without cofinal branches, there is a proper and $\aleph_2$-preserving forcing notion with…
This expository paper, aimed at the reader without much background in set theory or logic, gives an overview of Cohen's proof (via forcing) of the independence of the continuum hypothesis. It emphasizes the broad outlines and the intuitive…
We develop the theory of the forcing with trees and creatures for an inaccessible lambda continuing Ros{\l}anowski and Shelah math.LO/9807172, math.LO/9909115. To make a real use of these forcing notions (that is to iterate them without…
Let $T^*$ be an almost Suslin tree, that is, an Aronszajn tree with no stationary antichains. Krueger introduced a forcing axiom, $\mathrm{PFA}(T^*)$, for the class of proper forcings that preserve that $T^*$ is almost Suslin. He showed…
We study a family of variants of Jensen's\emph{subcomplete forcing axiom}, $\mathsf{SCFA}$ and \emph{subproper forcing axiom}, $\mathsf{SubPFA}$. Using these we develop a general technique for proving non-implications of $\mathsf{SCFA}$,…
We present a notion of $\Delta$-stability and stability filtration in arbitrary categories which is equivalent to the existence of Harder-Narasimhan (HN) sequences on objects. Indeed it is equivalent to the existence of a zero morphism, a…
We provide decidability and undecidability results on the model-checking problem for infinite tree structures. These tree structures are built from sequences of elements of infinite relational structures. More precisely, we deal with the…
We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice…
The basic result of this note is a statement about the existence of families of partitions of the set of natural numbers with some favourable properties, the n-optimal matrices of partitions. We use this to improve a decomposition result…
We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a $\boldsymbol{\Sigma}^1_2$ m.i.f. is equivalent to the existence of a $\boldsymbol{\Pi}^1_1$ m.i.f., (b) in the Cohen model,…
We prove that every {finitely generated residually finite}-by-sofic group satisfies Kaplansky's direct and stable finiteness conjectures with respect to all noetherian rings. We use this result to provide countably many new examples of…
Non-wellfounded proof systems impose a global condition called the global trace condition (GTC) on a derivation tree to ensure soundness. Providing a categorical characterisation of the GTC that guarantees soundness remains challenging due…
We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a…
We study a size-structured model proposed in [1] C. Barril, \`A. Calsina, O. Diekmann, J. Z. Farkas, On competition through growth reduction, e-print arXiv:2303.02981, to describe the dynamics of trees growth in the forest. Our approach to…
Phylogenetically decisive collections of taxon sets have the property that if trees are chosen for each of their elements, as long as these trees are compatible, the resulting supertree is unique. This means that as long as the trees…
We lay the ground for an Isabelle/ZF formalization of Cohen's technique of forcing. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize the definition of forcing notions as…
It is proven that the identity component of the group preserving the leaves of a generalized foliation is perfect. This shows that a well-known simplicity theorem on the diffeomorphism group extends to the nontransitive case.