Related papers: Totalities of Infinite Sets
In this article we consider alternative definitions-descriptions of a set being Infinite within the primitive Axiomatic System of Zermelo.
An algorith to count, or alternatively generate, all k-element transversals of a set system is presented and compared with three known methods. For special cases it works in output-linear time.
A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This approach concludes finally the problem of the…
We recover the rays in the tensor product of Hilbert spaces within a larger class of so called `states of compoundness', structured as a complete lattice with the `state of separation' as its top element. At the base of the construction…
This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are…
Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial…
In this paper, we have established boundaries of cardinal numbers of nonempty sets in finite non-$T_1$ topological spaces using interval analysis. For a finite set with known cardinality, we give interval estimations based on the closure…
A set $A$ of integers is called total if there is an algorithm which, given an enumeration of $A$, enumerates the complement of $A$, and called cototal if there is an algorithm which, given an enumeration of the complement of $A$,…
This paper enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: {\sf All $x$ are $y$} and {\sf Some $x$ are…
We investigate the theory of finite observables, i.e., resolutions of the finite-dimensional identity by means of positive operators, that have a physical interpretation in terms of measurement schemes. We focus on extremal and rank-one…
For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some…
This paper presents a novel approach to constructing finite generating sets for infinitely generated ideals. By integrating algebraic and computational techniques, we provide a method to identify finite generators, demonstrated through…
Discussions surrounding the nature of the infinite in mathematics have been underway for two millennia. Mathematicians, philosophers, and theologians have all taken part. The basic question has been whether the infinite exists only in…
We define and study Noetherian topologies for spaces of infinite sets, and infinite words. In each case, we also obtain S-representations, namely, computable presentations of the sobrifications of those spaces.
The paper puts into discussion the concept of universality, in particular for structures not of the power of Turing computability. The question arises if for such structures a universal structure of the same kind exists or not. For that the…
The notions of null-sets and nullity are present in all discourses of mathematics. They are based on the dual-pair of notions of "almost-every" and "almost none". A notion of nullity corresponds to a choice of subsets that one interprets as…
Quantum states that remain separable (i.e., not entangled) under any global unitary transformation are known as absolutely separable and form a convex set. Despite extensive efforts, the complete characterization of this set remains largely…
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…
Classical set theory constructs the continuum via the power set P(N), thereby postulating an uncountable totality. However, constructive and computability-based approaches reveal that no formal system with countable syntax can generate all…
A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid.…