Axiomatic Set Theory Research Articles

Up with Categories, Down with Sets; Out with Categories, In with Sets!

Abstract Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for ‘looking down’ or ‘in’ at subsets and the category-theoretic approach is the most practical for ‘looking up’ or ‘out’ at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory.

Russell Paradox Analysis: Descriptions and Logical Cases

The Russell Paradox is a classic problem in mathematical logic that challenges the underlying basics of set theory and the concept of infinity. This study provides a comprehensive analysis of the paradox, including its history, development, and recent research. The paradox arises from the set whose members are sets not containing themselves, which results in being contradictory. Bertrand Russell and Ernst Zermelo proposed early solutions, while the axiomatic set theory provided a rigorous foundation for mathematics. Recent studies suggest alternative solutions, including distinguishing between object-language and meta-language and introducing a new concept called a "Russell-class." This paper summarizes the history of the paradox, summarizes recent studies, and presents a framework for understanding and evaluating solutions. According to the analysis, it aims to deepen the understanding of the paradoxical situation and why it is of enormous importance for the underlying basics of mathematics. This paradox remains a subject of intense study and continues to inspire the development of various theories and frameworks to resolve it. Overall, these results shed light on guiding further exploration of the paradoxical situation and its enormous importance for the underlying basics of mathematics. They also present a framework for understanding and evaluating solutions, with the goal of deepening our understanding of the paradox and inspiring the development of various theories and frameworks to resolve the issue.

Locally compact, ω1-compact spaces

An ω1-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, ω1-compact space is σ-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties.Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, ω1-compact space of cardinality ℵ1 is σ-countably compact. Whether ℵ1 can be replaced with ℵ2 is a difficult unsolved problem. Modulo large cardinals, it is also ZFC-independent whether every hereditarily normal, or every monotonically normal, locally compact, ω1-compact space is σ-countably compact.As a result, it is also ZFC-independent whether there is a locally compact, ω1-compact Dowker space of cardinality ℵ1, or one that does not contain both an uncountable closed discrete subspace and a copy of the ordinal space ω1.Set theoretic tools used for the consistency results include the existence of a Souslin tree, the Proper Forcing Axiom (PFA), and models generically referred to as “MM(S)[S]”. Most of the work is done by the P-Ideal Dichotomy (PID) axiom, which holds in the latter two cases, and which requires no large cardinal axioms when directly applied to topological spaces of cardinality ℵ1, as it is in several theorems.

Axiom výběru a hypotéza kontinua – souvislosti a rozdíly

We compare two well-known set-theoretical statements, namely the axiom of choice and the continuum hypothesis, with regard to their historical development and formulation, as well as their consequences in mathematics. It is known that both statements are independent from the other axioms of set theory (if they are consistent). The axiom of choice – despite initial controversies – is today almost universally accepted as an axiom. However, the status of the continuum hypothesis is more complex and no agreement has been found so far: both the continuum hypothesis and its negation (often as consequences of stronger statements) decide several mathematical problems differently, but in contrast with the axiom of choice it is not clear which of the two solutions should be the “correct” one (in the sense of an agreement within the community).

SOME CONSEQUENCES OF AND

AbstractStrong Turing Determinacy, or ${\mathrm {sTD}}$ , is the statement that for every set A of reals, if $\forall x\exists y\geq _T x (y\in A)$ , then there is a pointed set $P\subseteq A$ . We prove the following consequences of Turing Determinacy ( ${\mathrm {TD}}$ ) and ${\mathrm {sTD}}$ over ${\mathrm {ZF}}$ —the Zermelo–Fraenkel axiomatic set theory without the Axiom of Choice: (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ —a weaker version of $\mathrm {DC}_{\mathbb {R}}$ .(2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every set of reals is measurable and has Baire property.(3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.(4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$ :(a)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $ , where $\mathrm {Dim_H}$ is the Hausdorff dimension.(b)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $ , where $\mathrm {Dim_P}$ is the packing dimension.

Reconciling transparency, low Δ0-complexity and axiomatic weakness in undecidability proofs

Abstract In a first-order theory $\varTheta $, the decision problem for a class of formulae $\varPhi $ is solvable if there is an algorithmic procedure that can assess whether or not the existential closure $\varphi ^{\exists }$ of $\varphi $ belongs to $\varTheta $, for any $\varphi \in \varPhi $. In 1988, Parlamento and Policriti already showed how to tailor arguments à la Gödel to a very weak axiomatic set theory, referring them to the class of $\varSigma _{1}$-formulae with $(\forall \exists \forall )_{0}$-matrix, i.e. existential closures of formulae that contain just restricted quantifiers of the forms $(\forall x \in y)$ and $(\exists x \in y)$ and are writable in prenex form with at most two alternations of restricted quantifiers (the outermost quantifier being a ‘$\forall $’). While revisiting their work, we show slightly less weak theories under which incompleteness for recursively axiomatizable extensions holds with respect to existential closures of $(\forall \exists )_{0}$-matrices, namely formulae with at most one alternation of restricted quantifiers.

Formal Verification of a Topological Spatial Relations Model for Geographic Information Systems in Coq

Geographic information systems have undergone rapid growth for decades. Topology has provided valuable modeling tools in the development of this field. Formal verification ofthe model of topological spatial relations can provide a reliable guarantee for the correctness of geographic information systems. We present a proof of the topological spatial relations model that has been formally verified in the Coq proof assistant. After an introduction to the formalization of the axiomatic set theory of Morse–Kelley, the formal description of the elementary concepts and properties of general topology is developed. The topological spatial relations between two sets are described by using the concept of the intersection value. Finally, we formally proved the topological spatial relations between two sets which are restricted to the regularly closed and the planar spatial regions. All the proof details are strictly completed in Coq, which shows that the correctness of the theoretical model for geographic information systems can be checked by a computer. This paper provides a novel method to verify the correctness of the topological spatial relations model. This work can also contribute to the creation and validation of various geological models and software.

Sharon Berry.A Logical Foundation for Potentialist Set Theory

This book offers a foundation for mathematics grounded in a collection of axioms for logical possibility in a first-order language. The offered foundation is argued to have various epistemological benefits, in particular as regards our justification for believing certain axioms of set theory, and more generally as regards the ‘access’ problem for mathematical objects. The foundation for mathematics is provided in stages. First, Berry argues for a ‘potentialist’ interpretation of standard ZF set theory in terms of some axioms for logical possibility that are argued to ‘seem clearly true’. Then, she argues for a Carnap-inspired account of the semantics of ordinary mathematical language in support of the idea that all mathematical discourse as usually pursued can be interpreted faithfully in terms of the foundation provided by her potentialist set theory. The end result, it is argued, is something approximating a logicist foundation for mathematics: standard mathematical truths are interpreted in terms of a pure logic of logical possibility.

SATURATED MODELS FOR THE WORKING MODEL THEORIST

AbstractWe put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

An elementary combinatorial theorem with an application to axiomatic set theory

An elementary combinatorial theorem with an application to axiomatic set theory

Metric Segments in Gromov-Hausdorff class

We study properties of metric segments in the class of all metric spaces considered up to an isometry, endowed with Gromov--Hausdorff distance. On the isometry classes of all compact metric spaces, the Gromov-Hausdorff distance is a metric. A metric segment is a class that consists of points lying between two given ones. By von Neumann--Bernays--Godel (NBG) axiomatic set theory, a proper class is a monster collection, e.g., the collection of all cardinal sets. We prove that any metric segment in the proper class of isometry classes of all metric spaces with the Gromov-Hausdorff distance is a proper class if the segment contains at least one metric space at positive distances from the segment endpoints. In addition, we show that the restriction of a non-degenerated metric segment to compact metric spaces is a non-compact set.

The Logical Study of Non-Well-Founded Set and Circulation Phenomenon

In recent years, since a great deal of circular phenomenon, there has been a furry of interest in them. To explain various circular phenomenon, the study of set theory extended well -founded sets to non-well-founded set. Based on this basis, the paper discusses the logical theoretical basis of circular phenomena. Non-well-founded set theory ZFA allows primitive existence. Primitive is an object that has no elements and is not a class in itself. It is based on the set theory ZFC after the axiom of foundation FA is removed, and the anti-basic axiom AFA is added to ZFC. ZFC here refers to ZF set theory with axiom of choice. According to axiomatic set theory, for ZFC's regular axioms, the set in its universe is a well set. If the regular axiom is removed, and the infinite decline is allowed to belong to the relational chain, then the non-well-founded set can be introduced. Firstly, this paper introduces the basic concept of non-well-founded set, the foundation axiom and the anti-founded axioms. Secondly, we dicusses the limit of the foundation axiom. Thirdly, we exhibit the history and present situation of the research on non-well-founded sets are briefly reviewed. Finally, the applications of non-well-founded sets in philosophy, linguistics, computer science, economics and many other fields is discussed. Because non-well-founded set theory will provide a better tool for dealing with circular phenomena naturally, it can be argued that circle is not vicious.

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

Bounded finite set theory

AbstractWe define an axiom schema for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic, , and use some of its basic model theory to establish some independence results for various axioms of set theory over . Then we ask: given a model M of , is there a model of whose ordinal arithmetic is isomorphic to M? We show that the answer is yes if .

Lines and Semi-Countably Differentiable Primes

Let l(u)⊃ |G|. A central problem in higher non-linear graph theoryis the construction of projective numbers. We show that Recent developments in axiomatic set theory [6] have raised the questionof whetherEis not dominated byl. On the other hand, the work in [6, 24] did not consider the hyper-real case.

Semisupervised Learning via Axiomatic Fuzzy Set Theory and SVM.

In this article, we present a semantic semisupervised learning (Semantic SSL) approach targeted at unifying two machine-learning paradigms in a mutually beneficial way, where the classical support vector machine (SVM) learns to reveal primitive logic facts from data, while axiomatic fuzzy set (AFS) theory is utilized to exploit semantic knowledge and correct the wrongly perceived facts for improving the machine-learning model. This novel semisupervised method can easily produce interpretable semantic descriptions to outline different categories by forming a fuzzy set with semantic explanations realized on the basis of the AFS theory. Besides, it is known that disagreement-based semisupervised learning (SSL) can be viewed as an excellent schema so that a co-training approach with SVM and the AFS theory can be utilized to improve the resulting learning performance. Furthermore, an evaluation index is used to prune descriptions to deliver promising performance. Compared with other semisupervised approaches, the proposed approach can build a structure to reflect data-distributed information with unlabeled data and labeled data, so that the hidden information embedded in both labeled and unlabeled data can be sufficiently utilized and can potentially be applied to achieve good descriptions of each category. Experimental results demonstrate that this approach can offer a concise, comprehensible, and precise SSL frame, which strikes a balance between the interpretability and the accuracy.

A Formal System of Axiomatic Set Theory in Coq

Formal verification technology has been widely applied in the fields of mathematics and computer science. The formalization of fundamental mathematical theories is particularly essential. Axiomatic set theory is a foundational system of mathematics and has important applications in computer science. Most of the basic concepts and theories in computer science are described and demonstrated in terms of set theory. In this paper, we present a formal system of axiomatic set theory based on the Coq proof assistant. The axiomatic system used in the formal system refers to Morse-Kelley set theory which is a relatively complete and concise axiomatic set theory. In this formal system, we complete the formalization of the basic definitions of sets, functions, ordinal numbers, and cardinal numbers and prove the most commonly used theorems in Coq. Moreover, the non-negative integers are defined, and Peano’s postulates are proved as theorems. According to the axiom of choice, we also present formal proofs of the Hausdorff maximal principle and Schroeder-Bernstein theorem. The whole formalization of the system includes eight axioms, one axiom schema, 62 definitions, and 148 corollaries or theorems. The “axiomatic set theory” formal system is free from the more apparent paradoxes, and a complete axiomatic system is constructed through it. It is designed to give a foundation for mathematics quickly and naturally. On the basis of the system, we can prove many famous mathematical theorems and quickly formalize the theories of topology, modern algebra, data structure, database, artificial intelligence, and so on. It will become an essential theoretical basis for mathematics, computer science, philosophy, and other disciplines.

Two axioms of set theory with applications to logic

Two axioms of set theory with applications to logic

The Semantic Classification Approach Base on Neural Networks

This paper addresses the classification problem, and a semantic classification approach using neural networks is proposed. The approach embeds the theoretical findings of the axiomatic fuzzy set theory in neural networks. Complex concepts are extracted by neural networks, which means that the class description is formed analytically rather than by tuning parameters of constraint conditions. The experiments are carried out on five benchmark datasets and compared results with five other neural network-based classifiers. The experimental results show that the proposed approach produces high classification accuracy and has a better explanation.

Research on Set Theory Based on Paraconsistent Logic

Different from ZF axiomatic set theory, the paraconsistent set theory has changed the basic logic of set theory and selected paraconsistent logic which can accommodate or deal with contradictions, it effectively avoids the whole theory falling into a non-trivial dilemma when there are contradictions in set theory. In this paper, we first review the history and current situation of the praconsistent set theory; then, we give three kinds of paraconsistent logic which can be used to construct the praconsistent set theory among many kinds of paraconsistent logics. And then, we analyze the differences of methods of the paraconsistent set theory with strong or weak structure of paraconsistent logic and get different paraconsistent set theory. Finally, we verify that paraconsistent set theory is a new method to solve the paradox of set theor. The development of paraconsistent set theory can solve the difficulties in the development of set theory in a unique way, which is not only the extension of the application of paraconsistent logic, but also the new form and new trend of the development of set theory.