Related papers: A Unified Approach to Algebraic Set Theory
Alternative set theory (AST) may be suitable for the ones who try to capture objects or phenomenons with some kind of indefiniteness of a border. While AST provides various notions for advanced mathematical studies, correspondence of them…
Axiomatic set theory is almost universally accepted as the basic theory which provides the foundations of mathematics, and in which the whole of present day mathematics can be developed. As such, it is the most natural framework for…
This is the second in a series of papers on the relation between algebraic set theory and predicative formal systems. In part I, we introduced the notion of a predicative category of small maps and obtained the result that such categories…
The construction of first-order logic and set theory gives rise to apparent circularities of mutual dependence, making it unclear which can act as a self-contained starting point in the foundation of mathematics. In this paper, we carry out…
This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative…
This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on Realisability and the other on Sheaf Models in Algebraic Set Theory.
The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…
CZF is a system of set theory which, over classical logic, is equivalent to ZF, while over intuitionistic logic, it has a well-known constructive type-theoretic interpretation. This article introduces a simpler, intuitive family of…
Approximation Fixpoint Theory (AFT) is an algebraic framework designed to study the semantics of non-monotonic logics. Despite its success, AFT is not readily applicable to higher-order definitions. To solve such an issue, we devise a…
We introduce and investigate the concept of Stratified Algebra, a new algebraic framework equipped with a layer-based structure on a vector space. We formalize a set of axioms governing intra-layer and inter-layer interactions, study their…
Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three…
Sets with atoms serve as an alternative to ZFC foundations for mathematics, where some infinite, though highly symmetric sets, behave in a finitistic way. Therefore, one can try to carry over analysis of the classical algorithms from finite…
The process of cognition is analysed to adjust the set theory to physical description. Postulates and basic definitions are revised. The specific sets of predicates, called presets, corresponding to the physical objects identified by an…
It is well-known that a finite axiomatization of Zermelo-Fraenkel set theory (ZF) is not possible in the same first-order language. In this note we show that a finite axiomatization is possible if we extent the language of ZF with the new…
In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for identity and set-membership. Although, these variations coincide with the…
Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a…
Association schemes on triples (ASTs) are ternary analogues of classical association schemes, whose relations and adjacency algebras are ternary instead of binary. We provide a survey of the current progress in the study of ASTs,…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…