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.

Object Relations as "Whole General Effect" in Management

Introduction. The socio-economic structure and the structure of economic production in Ukraine are get increasingly subjected to the consumption of goods and services. Problem Statement. In the outlined conditions, the problem is that in most cases, the economy has been already controlling the human being, not vice versa. Purpose. The development of a systematic approach to the problem of revealing the essence of the relationship between the “subject” of economic space management, the human being, and the “object” of management, the economic space, in order to find ways to return to human-centered socio-economic structure of the country. Materials and Methods. For the first time, a new concept of “social effect” has been introduced. It is an analog of the well-known lexicographic effect proposed at the beginning of the 21st century by Full Member of the NAS of Ukraine V. Shirokov, which may be regarded as a phenomenological framework of the theory of complexity and the corresponding specific theory of economic information, on the one hand. On the other hand, the complexity theory, the Kolmogorov information, and the Levenheim-Skolem theorem may be considered formal correlates of the lexicographic effects in economic systems. Results. This approach is considered system-creating to describe the holistic processes of relations of economic systems of five levels with the phenomenal property of self-compensation of complexity. The separation of structural, substantive, and subjective properties, as well as the relationships between them gives the analyzed economic reality the property of being a system. Conclusions. Based on the above assumptions and general theoretical and informational ideas about socio-economic systems of five levels as formal correlates of the lexicographic effect, it has been proposed to coordinatize and to unify information in the economic space of these management systems as a basis for establishing the “source — form — content” equilibrium in accordance with the rule of "common goals".

A model-theoretic approach to descriptive general frames: the van Benthem characterization theorem

The celebrated van Benthem characterization theorem states that on Kripke structures modal logic is the bisimulation-invariant fragment of first-order logic. In this paper, we prove an analogue of the van Benthem characterization theorem for models based on descriptive general frames. This is an important class of general frames for which every modal logic is complete. These frames can be represented as Stone spaces equipped with a ‘continuous’ binary relation. The proof of our theorem generalizes Rosen’s proof of the van Benthem theorem for finite frames and uses as an essential technique a new notion of descriptive unravelling. We also develop a basic model theory for descriptive general frames and show that in many ways it behaves like the model theory of finite structures. In particular, we prove the failure of the compactness theorem, of the Beth definability theorem, of the Craig interpolation theorem and of the upward Löwenheim–Skolem theorem.1

Object Relations as “Whole General Effect” in Management

Introduction. The socio-economic structure and the structure of economic production in Ukraine are get increasingly subjected to the consumption of goods and services. Problem Statement. In the outlined conditions, the problem is that in most cases, the economy has been already controlling the human being, not vice versa. Purpose. The development of a systematic approach to the problem of revealing the essence of the relationship between the “subject” of economic space management, the human being, and the “object” of management, the economic space, in order to find ways to return to human-centered socio-economic structure of the country. Materials and Methods. For the first time, a new concept of “social effect” has been introduced. It is an analog of the well-known lexicographic effect proposed at the beginning of the 21st century by Full Member of the NAS of Ukraine V. Shirokov, which may be regarded as a phenomenological framework of the theory of complexity and the corresponding specific theory of economic information, on the one hand. On the other hand, the complexity theory, the Kolmogorov information, and the Levenheim-Skolem theorem may be considered formal correlates of the lexicographic effects in economic systems. Results. This approach is considered system-creating to describe the holistic processes of relations of economic systems of five levels with the phenomenal property of self-compensation of complexity. The separation of structural, substantive, and subjective properties, as well as the relationships between them gives the analyzed economic reality the property of being a system. Conclusions. Based on the above assumptions and general theoretical and informational ideas about socio-economic systems of five levels as formal correlates of the lexicographic effect, it has been proposed to coordinatize and to unify information in the economic space of these management systems as a basis for establishing the “source — form — content” equilibrium in accordance with the rule of "common goals".

Categoricity and the natural numbers

This chapter focuses on modelists who want to pin down the isomorphism type of the natural numbers. This aim immediately runs into two technical barriers: the Compactness Theorem and the Löwenheim-Skolem Theorem (the latter is proven in the appendix to this chapter). These results show that no first-order theory with an infinite model can be categorical; all such theories have non-standard models. Other logics, such as second-order logic with its full semantics, are not so expressively limited. Indeed, Dedekind's Categoricity Theorem tells us that all full models of the Peano axioms are isomorphic. However, it is a subtle philosophical question, whether one is entitled to invoke the full semantics for second-order logic — there are at least four distinct attitudes which one can adopt to these categoricity result — but moderate modelists are unable to invoke the full semantics, or indeed any other logic with a categorical theory of arithmetic.

Additive maps having a generalized derivation expansion

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.