Related papers: Exhaustible sets in higher-type computation

A subset of a topological space is constructible if it is a finite Boolean combination of closed sets. We prove that every NTP$_2$ expansion of $(\mathbb{R},<,+)$ by constructible sets defines only constructible sets, and that definable…

Four constructions result from a desire to create enhancements to Cantor's infinite real set cardinality. Each continues to keep Cantor's cardinality formulation in place while providing new comparisons of arbitrary infinite sets. To…

We systematically investigate the complexity of model checking the existential positive fragment of first-order logic. In particular, for a set of existential positive sentences, we consider model checking where the sentence is restricted…

Below, by space we mean a separable metrizable zero-dimensional space. It is studied when the space can be embedded in a Cantor set while maintaining the algebraic structure. Main results of the work: every space is an open retract of a…

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…

This paper argues that mathematical objects are constructions and that constructions introduce a flexibility in the ways that mathematical objects are represented (as sets of binary sequences for example) and presented (in a particular…

Given a hereditarily meager ideal $\mathcal{I}$ on a countable set $X$ we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology $\tau^\mathcal{I}$ on $X$ such that $\tau^\mathcal{I}\cap…

We provide a characterization of when a countably infinite set of finite sets contains an infinite sunflower. We also show that the collection of such sets is Turing equivalent to the set of programs such that whenever the program converges…

We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…

We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic…

Given an uncountable cardinal $\kappa$, we consider the question of whether subsets of the power set of $\kappa$ that are usually constructed with the help of the Axiom of Choice are definable by $\Sigma_1$-formulas that only use the…

An archetypal problem discussed in computer science is the problem of searching for a given number in a given set of numbers. Other than sequential search, the classic solution is to sort the list of numbers and then apply binary search.…

A denumerable cellular family of a topological space $\mathbf{X}$ is an infinitely countable collection of pairwise disjoint non-empty open sets of $ \mathbf{X}$. It is proved that the following statements are equivalent in $\mathbf{ZF}$:…

We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth…

This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the category-theoretic version of the classical area of algebraic semantics. The…

We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the $\Sigma_\beta$ hierarchy. We focus on linear orderings. We show that at the $\Sigma_1$ level all linear…

We call a subset of an ordinal $\lambda$ recognizable if it is the unique subset $x$ of $\lambda$ for which some Turing machine with ordinal time and tape, which halts for all subsets of $\lambda$ as input, halts with the final state $0$.…

We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit,…

Let $X$ be a topological vector space of complex-valued sequences and $Y$ be a subset of $X$. We provide conditions for $X \setminus Y \cup \{0\}$ to contain uncountably infinitely many linearly independent dense vector subspaces of $X$. We…

Motivated by the theory of locally definable groups, we study the theory of $K$-vector spaces with a predicate for the union $X$ of an infinite family of independent subspaces. We show that if $K$ is infinite then the theory is complete and…