Related papers: $omega_{1}$ under $Pi_{1}$-Collection

The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality…

We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise…

We study semantic and syntactic properties of spherical orders and their elementary theories, including finite and dense orders and their theories. It is shown that theories of dense $n$-spherical orders are countably categorical and…

Let $\omega$ be a differential 1-form defining an algebraic foliation of codimension 1 in projective space. In this article we use commutative algebra to study the singular locus of $\omega$ through its ideal of definition. Then, we expose…

In this article, we deal with several notions in dynamical systems. Firstly, we prove that both closure function and orbital function are idempotent on set-valued dynamical systems. And we show that the compact limit set of a connected set…

We prove a counting theorem concerning the number of lattice points for the dual lattices of weakly admissible lattices in an inhomogeneously expanding box, which generalises a counting theorem of Skriganov. The error term is expressed in…

In this paper we consider big Ramsey degrees of finite chains in countable ordinals. We prove that a countable ordinal has finite big Ramsey degrees if and only if it is smaller than $\omega^\omega$. Big Ramsey degrees of finite chains in…

The countable condensation on a linear order $L$ is the equivalence relation $\sim_\omega$ defined by declaring $x \sim_\omega y$ when the set of points between $x$ and $y$ is countable. We characterize the linear orders $L$ that condense…

For every natural number $n$, there exist finitely presented groups with residual finiteness depths $\omega\cdot n$ and $\omega\cdot n + 1$. The ordinals that arise as the residual finiteness depth of a finitely generated group…

The first part of this article deals with theorems on uniqueness in law for \sigma-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two…

We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation…

We develop a theory of k-partitions of the set of infinite words recognizable by classes of finite automata. The theory enables to complete proofs of existing results about topological classifications of the (aperiodic) omega-regular…

This article expands our work in [Ca16]. By its reliance on Turing computability, the classical theory of effectivity, along with effective reducibility and Weihrauch reducibility, is only applicable to objects that are either countable or…

A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal-Katona theorem. A bound on non-consecutive clique numbers is also proven.

We isolate several classes of stationary sets of kappa^omega and investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.

We study a strengthening of Bounded Martin's Maximum which asserts that if a \Sigma_1 fact holds of \omega_2^V in a stationary set preserving extension then it holds in V for a stationary set of ordinals less than \omega_2. We show that…

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by…

We consider arbitrary open sets $\Omega$ in Euclidean space with finite Lebesgue measure, and obtain upper bounds for (i) the largest Courant-sharp Dirichlet eigenvalue of $\Omega$, (ii) the number of Courant-sharp Dirichlet eigenvalues of…

We establish the decidability of the $\Sigma_2$ theory of $\mathscr{D}_h(\leq_h \mathcal{O})$, the hyperarithmetic degrees below Kleene's $\mathcal{O}$, in the language of uppersemilattices with least and greatest element. This requires a…

We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a $\pmb\Pi_\omega^0$-complete set of models. In…