The Axiom of Projective Determinacy implies the existence of a universal $\utilde{\Pi}^{1}_{n}\setminus\utilde{\Delta}^{1}_{n}$ set for every $n \geq 1$. Assuming $\text{\upshape MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\utilde{\Pi}^{1}_{1}\setminus\utilde{\Delta}^{1}_{1}$ set. In ZFC there is a universal $\utilde{\Pi}^{0}_{\alpha}\setminus\utilde{\Delta}^{0}_{\alpha}$ set for every $\alpha$.

In this paper we summarize some results about sets in Frege structures. The resulting set theory is discussed with respect to its historical and philosophical significance. This includes the treatment of diagonalization in the presence of a universal set.

Here is a template for introducing mathematical objects: “Objects are found in stages. For every stage S: (1) for any things found before S, you find at S the bland set whose members are exactly those things; (2) for anything, x, which was found before S, you find at S the result of tapping x with any magic wand (provided that the result is not itself a bland set); you find nothing else at S.” -/- This Template has rich applications, (...) it realizes John Conway’s (1976) Mathematicians’ Liberation Movement, and it generalizes a lovely idea due to Alonzo Church (1974). -/- Some parts of this Template are familiar: the bit about bland sets is just the ordinary story we tell about the cumulative iterative conception of set. But this talk of “magic wands” is new and different. Moreover, parts of the Template are left unspecified. -/- This under-specification is deliberate: we want to be able to flesh out the Template in umpteen different ways. The Main Theorem of the paper is that any loosely constructive way of fleshing out the Template is synonymous with a ZF-like theory. (shrink)

In this note is proved the following:Theorem.Iƒ A × B is universal and one oƒ A, B is r.e. then one of A, B is universal.Letα, τbe 1-argument recursive functions such thatxgoes to, τ) is a map of the natural numbers onto all ordered pairs of natural numbers. A set A of natural numbers is calleduniversalif every r.e. set is reducible to A; A × B is calleduniversalif the set.

The impact of digital devices and the Internet has generated various changes at social, political, and economic levels, the repercussion of which is a great challenge characterized by the changing and globalized nature of today's society. This demands the development of new skills and new learning models in relation to information and communication technologies. Universities must respond to these social demands in the training of their future professionals. This paper aims to analyze the empirical evidence provided by international studies in (...) the last eleven years, related to the digital literacy of university students, including those pursuing degrees related to the field of education. Our findings highlight the fact that the digital literacy that is offered in universities to graduate/postgraduate students, in addition to treating digital literacy as a central theme, also focuses on perceived and developed self-efficacy. This is done by strengthening competencies related to digital writing and reading, the use of databases, the digital design of content and materials, and the skills to edit, publish or share them on the web, or applications aimed at treating digital literacy as emerging pedagogies and educational innovation. Secondly, we found studies related to digital competencies and use of the Internet, social networks, web 2.0, or the treatment of digital risks and their relationship with digital literacy. Thirdly, we found works that, in addition to focusing on digital literacy, also focused on different psychological constructs such as motivation, commitment, attitudes, or satisfaction.Systematic review registration:https://www.scopus.com/home.uri; https://www.recursoscientificos.fecyt.es/. (shrink)

ABSTRACT Most university students have engaged in some form of academic dishonesty. These actions can have detrimental consequences for the student, the university, and society at large. It is important to understand factors that contribute to academic dishonesty as well as to identify potential predictors of this behavior. This study employed an experimental design with 361 undergraduate students in a laboratory setting. Deception was used during the experiment to determine the impact of peer influence, personality, and an honor code on (...) cheating behavior. Results revealed that peer influence had the largest impact on cheating behavior. In addition, personality, specifically, individuals who scored high on the openness trait were significantly more likely to cheat than those who scored lower on this trait. Implications of these findings for addressing academic dishonesty in higher education are provided. (shrink)

Zermelo once wrote that he had anticipated Russell's contradiction of the set of all sets that are not members of themselves. Is this sufficient for having anticipated Russell's Paradox — the paradox that revealed the untenability of the logical notion of a set as an extension? This paper argues that it is not sufficient and offers criteria that are necessary and sufficient for having discovered Russell's Paradox. It is shown that there is ample evidence that Russell satisfied the criteria (...) and that Zermelo did not. (shrink)

Let X, Y be Polish spaces,,. We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set is denoted by, where. The following results are shown: If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and. If A is a set universal for all countable subsets (...) of Y, then is a equivalence relation, and and ; if, then ; if every set is Lebesgue measurable or has the Baire property, then. for, if every set has the Baire property, and E is any equivalence relation, then. (shrink)

This study explored the experiences of the newly-hired faculty in the university to acquire an in-depth and profound understanding of these experiences. It involved eight purposively chosen faculty in a semi-structured interview and focus group discussion documented in voice and video recorders. Using Colaizzi’s seven step-approach to data analysis, results revealed three paradoxical themes: the joys and struggles of teaching, successes and pains of learning, and the privileges and challenges of serving. On the one hand, the joys and struggles of (...) teaching were brought about by their adjustments in instruction. On the other hand, successes and pains of learning centered on professional growth. Finally, the privileges and challenges of serving incorporated the experiences in research, production, and extension. Generally, these teachers have experienced ups and downs in their first two years of teaching at a higher education institution. On the basis of study findings, implications and recommendations are discussed. (shrink)

The concept of organizational justice is important to understanding and predicting organizational behavior. A significant development in the research literature has been the separation of distributive and procedural justice. While much of the research has focused on negative outcomes, this research attempted to verify the presence of both forms of justice in the context of positive outcomes. Subjects completed an instrument designed to measure their perceptions of distributive and procedural justice. The subjects also reported their satisfaction and sense of fairness (...) with their salary increases, their belief that the procedures to award the increases had been followed, and their level of information and agreement regarding the salary program. These measures, along with size of salary increase and gender were examined to determine their impact on the subjects' perceived level of justice. The data support the existence of the two distinct forms of justice, but suggest that procedural justice may, in turn, branch out into two aspects. One category involves being informed, and a second appears to deal with acceptance of procedures. A series of relationships are then considered. Significant gender effects were non-existent. (shrink)

The book discusses the fate of universality and a universal set in several set theories.

As health policy-makers around the world seek to make progress towards universal health coverage, they must navigate between two important ethical imperatives: to set national spending priorities fairly and efficiently; and to safeguard the right to health. These imperatives can conflict, leading some to conclude that rights-based approaches present a disruptive influence on health policy, hindering states’ efforts to set priorities fairly and efficiently. Here, we challenge this perception. We argue first that these points of tension stem largely from (...) inadequate interpretations of the aims of priority setting as well as the right to health. We then discuss various ways in which the right to health complements traditional concerns of priority setting and vice versa. Finally, we set out a three-step process by which policy-makers may navigate the ethical and legal considerations at play. (shrink)

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.

Category theory is often treated as an algebraic foundation for mathematics, and the widely known algebraization of ZF set theory in terms of this discipline is referenced as “categorical set theory” or “set theory for category theory”. The method of algebraization used in this theory has not been formulated in terms of universal algebra so far. In current paper, a _universal algebraic_ method, i.e. one formulated in terms of universal algebra, is presented and used for algebraization of a (...) ground mereological system described as “quantitative” (QMS) and of the standard set theories built upon it. The QMS is algebraized by using abelian generalized groups (AGGs) generalizing both abelian groups and Boolean algebras. To algebraize set theory, the AGG are enriched with operators to produce _extensional algebras_ (EA) of Tarskian “Boolean algebras with operators” type. EAs explicate the intuition of set theory’ universes (of discourse). The axiomatic set theory presented in this paper, named _extension theory_, previously went through an intuitive phase of development during its use by the author in computer science projects, where it served as the basis for development of a foundational data science framework and an extensional model of natural languages also outlined here. (shrink)

In this article ‡ we show how the universe of HST, Hrbaček set theory admits a system of subuniverses which keep the Replacement, model Power set and Choice, and also keep as much of Saturation as it is necessary. This gives sufficient tools to develop the most complicated topics in nonstandard analysis, such as Loeb measures.

Let n ≥ 3 be an integer. We show that it is consistent (relative to the consistency of n - 2 strong cardinals) that every $\Sigma_n^1-set$ of reals is universally Baire yet there is a (lightface) projective well-ordering of the reals. The proof uses "David's trick" in the presence of inner models with strong cardinals.

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.We discuss here the corresponding situation for fuzzy set theory. Our emphasis will be on various approaches toward (more or less naively formed) universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.

This paper is motivated by the search for a natural and deductively powerful extension of classical set theory. A theory of properties U is developed, based on a system of relevant logic related to RQ. In U the set {a, b, c,...} is identified with the property [x: x=a x=b x=c...]. The universe of all sets V, is identified with the property of being a hereditary set. The main result is that relevant implication collapses to material implication for sentences (...) with quantifiers restricted to V. This demonstrates the naturalness of the system. However, an aparent lack of deductive power leads to the conclusion that the best extension of classical set theory is to be found in intensional theories with the unrestricted comprehension schema based on weak relevant logics. The author has obtained similar collapses of to for these systems. (shrink)

Let n ≥ 3 be an integer. We show that it is consistent that every σ1n-set of reals is universally Baire yet there is a projective well-ordering of the reals. The proof uses “David’s trick” in the presence of inner models with strong cardinals.

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems (...) to necessitate the addition of subsets to V. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and analyse various tradeoffs in naturality that might be made. We conclude that the Universist has promising options for interpreting different forcing constructions. (shrink)

According to the iterative conception of sets, standardly formalized by ZFC, there is no set of all sets. But why is there no set of all sets? A simple-minded, though unpopular, “minimal” explanation for why there is no set of all sets is that the supposition that there is contradicts some axioms of ZFC. In this paper, I first explain the core complaint against the minimal explanation, and then argue against the two main alternative answers to (...) the guiding question. I conclude the paper by outlining a close alternative to the minimal explanation, the conception-based explanation, that avoids the core complaint against the minimal explanation. (shrink)

We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 211–243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of (...) the existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness. (shrink)