If a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results.

In this paper we prove that the preordering $\lesssim $ of provable implication over any recursively enumerable theory $T$ containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function $F$ for $\lesssim $. A recursive function $F$ is a density function if it computes, for $A$ and $B$ with $A\lnsim B$, an element $C$ such that $A\lnsim C\lnsim B$. The function is extensional if it preserves $T$-provable equivalence. Secondly, we prove a (...) general result that implies that, for extensions of elementary arithmetic, the ordering $\lesssim $ restricted to $\Sigma_{n}$-sentences is uniformly dense. In the last section we provide historical notes and background material. (shrink)

We show the non-arithmeticity of 1st order theories of lattices of Σ n sentences modulo provable equivalence in a formal theory, of diagonalizable algebras of a wider class of arithmetic theories than has been previously known, and of the lattice of degrees of interpretability over PA. The first two results are applications of Nies’ theorem on the non-arithmeticity of the 1st order theory of the lattice of r.e. ideals on any effectively dense r.e. Boolean algebra. The theorem on degrees of (...) interpretability relies on an adaptation of techniques leading to Nies’ theorem. (shrink)

We prove that the diagonalizable algebras of PA and ZF are not isomorphic.

Religious experience differs from the empirical experience of the subject by psychologicality, the transcendental vitality of understanding objective phenomena. The main criterion of a person's religious experience is his belief in the truth of the existing a priori and the interrelations of things and phenomena of the objective and subjective world revealed to him in personal experience. Faith is a sense of the interconnection between the subject and the object, which has an experienced transcendental character. Human experience in this respect (...) acts as a factor in the disclosure of depth and effectiveness: a subject-object relationship accepted on faith: the disclosure of the meaning and directions of its development relative to the object of faith; as a set of acts of self-knowledge in the context of the existence of the object and, only thanks to the object of faith, brought into a whole, meaningful existence. Thus, the self-knowledge of a person depends, first of all, on the choice of an object of faith for him, as the meaning of his development. (shrink)

Absract. The regulatory documents of the Ministry of Higher Education and Science set the task of forming competencies of students. Competences are understood as knowledge and skills. Knowledge is the world outlook component of science, and therefore the corresponding academic discipline, and skills are its methodological component. The world outlook refers to the system of views on the subject area of science, and the methodology is the system of intangible means of cognition and practical activities developed in this science and (...) presented in the relevant academic discipline. It is noted that the subject of study in logic is not only the particular structure of thoughts and processes of thinking, but also the types and characteristics of the original components of complex thoughts and processes of thinking, such as signs and terms. In the article on the example of logic the author's concept of the methodological component of the academic discipline is described. Attention is drawn to the importance of using in the process of cognition methodological standpoints created on the basis of world outlook, as well as to the need to take into account the types of knowledge - empirical and theoretical knowledge, problem, hypothesis, theory. Some other methodological tools of logic are given, for example, principles, technology. More well-known methodological tools are not considered. It is assumed that the article will cause interest among the representatives of other philosophical, and not only philosophical, academic disciplines, and a discussion of the problem of ideological and methodological components of these disciplines will continue. (shrink)

We prove that the first order theory of the fixed point algebra corresponding to an r.e. consistent theory containing arithmetic is hereditarily undecidable.

We present a decision procedure for the ∀∃ theory of the lattice of Σ1 sentences of Peano Arithmetic.

We present a decision procedure for the ∀∃ theory of the lattice of Σ1 sentences of Peano Arithmetic.

The technical literature actively discusses ideas for creating digital twins to solve a wide range of problems that arise for enterprises of various kinds. Advances in the areas of information technology and the development of the necessary software really make it possible to obtain important new results through the creation of digital twins. This paper proposes the concept of building a network of digital twins, used to solve a number of actual problems in complex telecommunication systems. Two applications are considered (...) examples of such tasks. The first application is the organization of information feedback for enterprises involved in the life cycle of a telecommunication network: “elaboration of modernization principles, development of equipment, design, construction, and operation.” The second application is monitoring traffic, including its atypical behavior, in order to avoid the recurrence of erroneous actions in case of emergency situations. Another important feature of the digital twin network is that it will become an effective tool for conducting interdisciplinary research. (shrink)

В монографии в широком историческом контексте охарактеризован процесс возникновения и развития сущностного наполнения принципа природосообразности в истории филисофско-педагогической мысли от античного периода по Новое время.

The presented enhanced version of Eriksen’s theorem defines an universal transform of the Foldy–Wouthuysen type and in any external static electromagnetic field reveals a discrete symmetry of Dirac’s equation, responsible for existence of a highly influential conserved quantum number—the charge index distinguishing two branches of DE spectrum. It launches the charge-index formalism obeying the charge-index conservation law. Via its unique ability to manipulate each spectrum branch independently, the CIF creates a perfect charge-symmetric architecture of Dirac’s quantum mechanics, which resolves all (...) the riddles of the standard DE theory. Besides the abstract CIF algebra, the paper discusses: the novel accurate charge-symmetric definition of the electric-current density; DE in the true-particle representation, where electrons and positrons coexist on equal footing; flawless “natural” scheme of second quantization; and new physical grounds for the Fermi–Dirac statistics. As a fundamental quantum law, the CICL originates from the kinetic-energy sign conservation and leads to a novel single-particle physics in strong-field situations. Prohibiting Klein’s tunneling in Klein’s zone via the CICL, the precise CIF algebra defines a new class of weakly singular DE solutions, strictly confined in the coordinate space and experiencing the total reflection from the potential barrier. (shrink)

The necessity to provide for a stable regional development as one of the main priorities of regional social and economic policies in the unstable world economics is justified. A wide classification of stability types in region development is considered and factors influencing the institutional stability are singled out.

One of the least investigated, which now exists in Ukraine is the Old Believers. Despite the fact that their communities have been operating in our territory for more than two centuries, researchers have long devised this topic with their attention. The main reason for this attitude to the study of the Old Believers should be considered political realities of the XX century, and they are known to have not contributed to the objective study of even traditional Ukrainian denominations. Therefore, right (...) now, with the proclamation by the state of equal treatment of all religions and non-interference in their affairs, the time has come to make up for the lost. Let us stop here only on one aspect of the problem - the activities of the Old Believers monasteries. (shrink)

Simone de Beauvoir is a representative of one of the leading philosophical schools in the middle of the 20th century. The article presents Beauvoir’s artistic method, applied in her novel The Mandarins, and examines the theoretical and biographical sources of the novel. The author demonstrates the place that the novel has in the Beauvoir’s literary and philosophical heritage and reveals the genre features of the work, introducing some special terms such as engaged, modern or philosophical novel and testimonial autobiographical project. (...) The article also analyzes the novel’s literary form and the binary structure of the narrative. The study of the main characters, who are Henri Perron, Anne Dubreuilh and her husband Robert, allows to give a couple of narrative lines. First of them is the inner line that opens the reflective, contemplative and intimate life of one of the main characters – Anne. The second one is the outer line that means that the reader receives the information about characters from the Henry’s actions. Basing on this structure, we draw a conclusion about the modifications in the genre of existential novel in the postwar years. The new themes can be found in the literature. Authors introduce to readers the certain social reality through the inner life of some characters – intellectuals, novelists or philosophers. The thesis about the inner transformation of the genre is proved on Beauvoir’ and Jean-Paul Sartre’s works and on the prewar works of Sartre and Albert Camus. Beauvoir’s new literary methods and plots, which are the logical development of her work, made her novel one of the pioneers in the postwar literature. (shrink)

The paper contains some observations on e-complete, precomplete, and uniformly finitely precomplete r. e. equivalence relations. Among these are a construction of a uniformly finitely precomplete r. e. equivalence which is neither e- nor precomplete, an extension of Lachlan's theorem that all precomplete r. e. equivalences are isomorphic, and a characterization of sets of fixed points of endomorphisms of uniformly finitely precomplete r. e. equivalences.

For over a century, Haeckel's Gastraea theory remained a dominant theory to explain the origin of multicellular animals. According to this theory, the animal ancestor was a blastula‐like colony of uniform cells that gradually evolved cell differentiation. Today, however, genes that typically control metazoan development, cell differentiation, cell‐to‐cell adhesion, and cell‐to‐matrix adhesion are found in various unicellular relatives of the Metazoa, which suggests the origin of the genetic programs of cell differentiation and adhesion in the root of the Opisthokonta. Multicellular (...) stages occurring in the complex life cycles of opisthokont protists (mesomycetozoeans and choanoflagellates) never resemble a blastula. Here, we discuss a more realistic scenario of transition to multicellularity through integration of pre‐existing transient cell types into the body of an early metazoon, which possessed a complex life cycle with a differentiated sedentary filter‐feeding trophic stage and a non‐feeding blastula‐like larva, the synzoospore. Choanoflagellates are considered as forms with secondarily simplified life cycles. (shrink)