The main result is that for sets ${S \subseteq \omega + 1}$ , the following are equivalent: The shuffle sum σ(S) is computable.The set S is a limit infimum set, i.e., there is a total computable function g(x, t) such that ${f(x) = \lim inf_t g(x, t)}$ enumerates S.The set S is a limitwise monotonic set relative to 0′, i.e., there is a total 0′-computable function ${\tilde{g}(x, t)}$ satisfying ${\tilde{g}(x, t) \leq \tilde{g}(x, t+1)}$ such that ${{\tilde{f}(x) = \lim_t \tilde{g}(x, t)}}$ (...) enumerates S.Other results discuss the relationship between these sets and the ${\Sigma^0_3}$ sets. (shrink)

We present the logic BLChang, an axiomatic extension of BL (see [23]) whose corresponding algebras form the smallest variety containing all the ordinal sums of perfect MV-chains. We will analyze this logic and the corresponding algebraic semantics in the propositional and in the first-order case. As we will see, moreover, the variety of BLChang-algebras will be strictly connected to the one generated by Chang’s MV-algebra (that is, the variety generated by all the perfect MV-algebras): we will also give some (...) new results concerning these last structures and their logic. (shrink)

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of (...) all n-element subsets for each natural number n ≥ 2. Moreover, series ∑β<αf are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic. (shrink)

In this highly original study, Nuccio Ordine uses the figure of the ass as a lens through which to focus on the thought and writings of the great Renaissance humanist philosopher Giordano Bruno.

“A little masterpiece of originality and clarity.”—George Steiner “A necessary book.”—Roberto Saviano “A wonderful little book that will delight you.”—François Busnel International Best Seller / Now in English for the First Time In this thought-provoking and extremely timely work, Nuccio Ordine convincingly argues for the utility of useless knowledge and against the contemporary fixation on utilitarianism—for the fundamental importance of the liberal arts and against the damage caused by their neglect. Inspired by the reflections of great philosophers and writers (e.g., (...) Plato, Dante, Montaigne, Shakespeare, Borges, and Calvino), Ordine reveals how the obsession for material goods and the cult of utility ultimately wither the spirit, jeopardizing not only schools and universities, art, and creativity, but also our most fundamental values—human dignity, love, and truth. Also included is Abraham Flexner’s 1939 essay “The Usefulness of Useless Knowledge,” which originally prompted Ordine to write this book. Flexner—a founder and the first director of the Institute for Advanced Study in Princeton—offers an impassioned defense of curiosity-driven research and learning. (shrink)

The home learning environment includes what parents do to stimulate children’s literacy and numeracy skills at home and their overall beliefs and attitudes about children’s learning. The home literacy and numeracy environments are two of the most widely discussed aspects of the home learning environment, and past studies have identified how socioeconomic status and parents’ own abilities and interest in these domains also play a part in shaping children’s learning experiences. However, these studies are mostly from the West, and there (...) has been little focus on the situation of homes in Asia, which captures a large geographical area and a wide diversity of social, ethnic, and linguistic groups. Therefore, this paper aims to review extant studies on the home literacy and numeracy environments that have been conducted in different parts of Asia, such as China, the Philippines, India, Iran, Turkey, and the United Arab Emirates. Specifically, we explore how parents in these places perceive their roles in children’s early literacy and numeracy development, the methods they regard as effective for promoting young children’s literacy and numeracy learning, and the frequency with which they engage their young children in different types of home literacy and numeracy activities. We also examine studies on the relationship of the home literacy and numeracy environment with young children’s developmental outcomes, and the effectiveness of parent training programs to improve the home literacy and numeracy environments in these contexts. By examining potential trends in findings obtained in different geographical areas, we can initially determine whether there are characteristics that are potentially unique to contexts in Asia. We propose future research directions that acknowledge the role of cultural values and social factors in shaping the home learning environment, and, by extension, in facilitating children’s early literacy and numeracy development. (shrink)

ABSTRACTWe examine Marx's critiques of language, politics, and capitalist political economy and show how these anticipated critical discourse and argumentation analysis and ‘cultural political economy’. Marx studied philology and rhetoric at university and applied their lessons critically. We illustrate this from three texts. The German Ideology critically explores language as practical consciousness, the division of manual and mental labor, the state, hegemony, intellectuals, and specific ideologies. The Eighteenth Brumaire studies the semantics and pragmatics of political language and how it represents (...) the class content of politics and contributes to social transformation. Capital deconstructs the categories of classical political economy and their constitutive role in capitalist social relations. This is one aspect of CPE. Capital also highlights the structural and agential aspects of these relations, their contradictory dynamic, and their crisis-prone character. We comment on this aspect too. This said, Marx held that social transformation is mediated through political imaginaries and highlighted the need for the proletariat to develop a ‘poetry’ of the future. We then consider the misleading ‘base-superstructure’ metaphor and note how, against the thrust of Marx's work, it tends to reify culture. The article concludes that Marx contributed to the critique of semiotic as well as political economy. (shrink)

According to Thomas Kuhn, the entrenchment of a paradigm, especially in the critical stages when facing anomalies, requires the further suppression of competing ideas. This essay addresses the unconscious entrenchment of European modernist aesthetics in the everyday, especially in the American suburbs of the 1950s, and its popular and cultural manifestations. Taking Levittown as a starting point, modernist architectural principles have since its construction radiated into the mass-housing market and materialized in housing development projects that have led to the rise (...) of suburbia and, more recently, the New Urbanism. Despite the conventional separations of modernism from the more criticized mass-housing development, the same principles of security, shelter, community, and utopia are present in each. Presented by some architects as a postmodernist aesthetic, the central tenets of modernism are still pursued and reimagined as timeless principles in the building of new housing communities, recycled and reconstituted to meet the dreams and desires of the public—a utopia that is both avant-garde and kitsch. (shrink)

Though the Random Utility Model (RUM) was conceived entirely in terms of ordinal utility, the apparatus through which it is widely practised exhibits properties of cardinal utility. The adoption of cardinal utility as a working operation of ordinal is perfectly valid, provided interpretations drawn from that operation remain faithful to ordinal utility. The article considers whether the latter requirement holds true for several measurements commonly derived from RUM. In particular it is found that measurements of consumer surplus change may depart (...) from ordinal utility, and exploit the cardinality inherent in the practical apparatus. (shrink)

Physical literacy has received considerable attention in the field of physical education and physical activity worldwide. According to recent studies, the level of physical fitness among Chinese university students is gradually decreasing. This study aims to examine the impact of the PF level on PA and PL, as well as the relationships among PF, PA, and PL, in Chinese university students. Participants comprised 798 university students in Chongqing, China. Participants completed the tests of vital capacity, cardiorespiratory fitness, muscular strength, and (...) flexibility, according to the National Physical Fitness Measurement Standards Manual, as well as questionnaires on PA and PL. The independent t-tests were conducted to examine sex differences in the evaluated variables, and the Pearson’s correlation between each PF test and PL attributes and PA was calculated according to sex. In addition, the independent t-tests were conducted to determine whether the PF level had an effect on PL attributes and PA at various intensities. Significant sex differences were found in the PF domains of vital capacity, muscular strength, and aerobic fitness, with higher scores in men than in women, but not in the PF domain of flexibility and total PL score. Furthermore, the PF domains of muscular strength and aerobic fitness were significantly and positively correlated with the PL attributes of confidence and physical competence in both men and women, while the PF domains of vital capacity and aerobic fitness were significantly and positively correlated with the PL attribute of motivation in men. In addition, PL was significantly and positively associated with cardiorespiratory fitness, vital capacity, muscular strength, and flexibility among participants in the fit group. These findings support advocating for increased participation in PA in university students and using PL as a tool to improve PF components. (shrink)

No research thus far has attempted to examine ethical decision- making in corporate entrepreneurial organizations. Results of such study would provide management executives with insights on what action, if any, is essential for achieving business ethics and corporate entrepreneurship simultaneously. This paper argues, theoretically, that the work characteristics, organizational characteristics, and some individual characteristics in a corporate entrepreneurial organization are conducive to ethical decisions. These characteristics help mitigate the adverse impact of the turbulent environments on ethical decision- making behavior. Based (...) on these arguments, a tentative model of ethical decison-making in corporate entrepreneurial organization is constructed. (shrink)

Many are calling for concrete mechanisms of oversight for health research involving artificial intelligence (AI). In response, institutional review boards (IRBs) are being turned to as a familiar model of governance. Here, we examine the IRB model as a form of ethics oversight for health research that uses AI. We consider the model's origins, analyze the challenges IRBs are facing in the contexts of both industry and academia, and offer concrete recommendations for how these committees might be adapted in order (...) to provide an effective mechanism of oversight for health‐related AI research. (shrink)

In this paper, we explore the effects of certain forbidden substructure conditions on preordered sets. In particular, we characterize in terms of these conditions those preordered sets which can be represented as the supremum of a well-ordered ascending chain of lowersets whose members are constructed by means of alternating applications of disjoint union and ordinal sums with chains. These decompositions are examples of ordinal decompositions in relatively normal lattices as introduced by Snodgrass, Tsinakis, and Hart. We conclude the paper (...) with an application to information systems. (shrink)

Aims: To examine the extent of social stigma toward diabetes among Singapore's multi-ethnic general population and determine whether this differs across socio-demographic sub-groups.Methods: Data for this study came from a nationwide cross-sectional study. A diabetes stigma questionnaire comprising Social Distance Scale and Negative Attitudes and Stereotyping Scale was administered to those respondents who had not been diagnosed with diabetes. Exploratory factor analysis was conducted to determine the dimensionality of the instruments and validated using confirmatory factor analysis. Multiple linear regression analysis (...) was conducted to examine associations between socio-demographic factors and measures of diabetes stigma.Results: In all, 2,895 participants were recruited from the general population giving a response rate of 66.2%. Factor analyses found that a one-factor model resulted in an acceptable fit for both stigma scales, which measured social distance and negative attitudes and stereotyping, respectively. Multiple linear regression analyses identified Indian ethnicity, higher personal income and having close friends or family members who had been diagnosed with diabetes to be significantly associated with lower social distance scores while those aged 50–64 years and those with secondary and vocational education were significantly associated with higher social distance scores. Those with a personal income of SG$2,000–3,999 and SGD $6,000 and above, and those with close friends or family members diagnosed with diabetes were significantly associated with lower negative attitudes and stereotyping scores. In contrast those aged 35 years and above, those with primary education and below, and those of Malay ethnicity were significantly associated with higher negative attitudes and stereotyping scores.Conclusions: The study found a relatively low level of stigma toward diabetes in the general population of Singapore, although some stigmatizing beliefs emerged. While greater knowledge of diabetes could reduce stigma, anti-stigma messaging should be incorporated into the “War on Diabetes” programme in Singapore. (shrink)

The essay discusses the presumption of one's singularity, the uniqueness of one's time, the picturesqueness of one actions, and the capacity of human beings, whether corporately or individually, to begin everything or indeed anything again from scratch. Such presumptions are indeed present in some varieties of contemporary fanaticism, but, more to the point, it is suggested that the feeling of doing something for the first time is the oldest feeling in the world.

The structure of ordinals of the form ωωβ for countable β is studied. The main result is:Theorem 1If β<ω1 is the sum of one or two indecomposable ordinals, thenωωβ→2. Also an example is given to show that α→2 need not imply α→2 for all n<ω.

We study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages. We show that such an iterated natural sum differs from the more usual transfinite ordinal sum only for a finite number of iteration steps. The iterated natural sum of a sequence of ordinals can be obtained as a mixed sum (in an order‐theoretical sense) of the ordinals in the sequence; in fact, it is the largest mixed sum which satisfies a (...) finiteness condition. We introduce other infinite natural sums which are invariant under permutations and show that all the sums under consideration coincide in the countable case. (shrink)

Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a, so that for every \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, i.e., two sets A, B, with A properly contained in B, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of ω; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On (...) the other hand, we also give a sufficient condition on a, that yields that there is a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal ω + ω; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n ∈ ω, or n = ω, and every notation a of a nonzero ordinal there exists a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of cardinality n, whose Rogers semilattice consists of exactly one element. (shrink)

Ordinal addition and multiplication can be extended in a natural way to all sets. I survey the structure of the sets under these operations. In particular, the natural partial ordering associated with addition of sets is shown to be a tree. This allows us to prove that any set has a unique representation as a sum of additively irreducible sets, and that the non-empty elements of any model of set theory can be partitioned into infinitely many submodels, each isomorphic to (...) the original model. Also any model of set theory has an isomorphic extension in which the empty set of the original model is non-empty. Among other results, the relations between the arithmetical operations and the transitive closure and the adductive hierarchy are elucidated. (shrink)

Pseudo-BCK-algebras are a non-commutative generalization of well-known BCK-algebras. The paper describes a situation when a linearly ordered pseudo-BCK-algebra is an ordinal sum of linearly ordered cone algebras. In addition, we present two identities giving such a possibility of the decomposition and axiomatize the residuation subreducts of representable pseudo-hoops and pseudo-BL-algebras.

BL-algebras are the Lindenbaum algebras for Hájek's Basic Logic, just as Boolean algebras correspond to the classical propositional calculus. The finite totally ordered BL-algebras are ordinal sums of MV-chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite BL-chain, and we use it to describe the injective and the weak injective members of these classes.

Is well-being the kind of thing that can be summed across individuals? This paper takes a measurement-theoretic approach to answering this question. To make sense of adding well-being, we would need to identify some natural "concatenation" operation on the bearers of well-being that satisfies the axioms of extensive measurement and can therefore be represented by the arithmetic operation of addition. I explore various proposals along these lines, involving the concatenation of segments within lives over time, of entire lives led alongside (...) one another or in sequence, and of evaluatively basic propositions via conjunction. All of these proposals turn out to carry highly controversial commitments about the good. I do not claim that these commitments are unacceptable. But they suggest that we cannot simply take for granted, as many philosophers do, that there is any such thing as the sum of well-being. (shrink)

Consider any subvariety of BL-algebras generated by a single BL-chain which is the ordinal sum of the standard MV-algebra on [0, 1] and a basic hoop H. We present a geometrical characterization of elements in the finitely generated free algebra of each of these subvarieties. In this characterization there is a clear insight of the role of the regular and dense elements of the generating chain. As an application, we analyze maximal and prime filters in the free algebra.

In [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ₁-elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ₁-elementarity as defined in [2]. This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see [6]. In the present paper we establish an effective order isomorphism between the classical and (...) the new system of ordinal notations using the results from [6] and [7]. Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization. (shrink)

Systems biologists often distance themselves from reductionist approaches and formulate their aim as understanding living systems “as a whole.” Yet, it is often unclear what kind of reductionism they have in mind, and in what sense their methodologies would offer a superior approach. To address these questions, we distinguish between two types of reductionism which we call “modular reductionism” and “bottom-up reductionism.” Much knowledge in molecular biology has been gained by decomposing living systems into functional modules or through detailed studies (...) of molecular processes. We ask whether systems biology provides novel ways to recompose these findings in the context of the system as a whole via computational simulations. As an example of computational integration of modules, we analyze the first whole-cell model of the bacterium M. genitalium. Secondly, we examine the attempt to recompose processes across different spatial scales via multi-scale cardiac models. Although these models rely on a number of idealizations and simplifying assumptions as well, we argue that they provide insight into the limitations of reductionist approaches. Whole-cell models can be used to discover properties arising at the interfaces of dynamically coupled processes within a biological system, thereby making more apparent what is lost through decomposition. Similarly, multi-scale modeling highlights the relevance of macroscale parameters and models and challenges the view that living systems can be understood “bottom-up.” Specifically, we point out that system-level properties constrain lower-scale processes. Thus, large-scale modeling reveals how living systems at the same time are more and less than the sum of the parts. (shrink)

This study aimed to investigate whether the ordinal judgments of high math-anxious and low math-anxious individuals differ. Two groups of 20 participants with extreme scores on the Shortened Mathematics Anxiety Rating Scale had to decide whether a triplet of numbers was presented in ascending order. Triplets could contain one-digit or two-digit numbers and be formed by consecutive numbers, numbers with a constant distance of two or three or numbers with variable distances between them. All these triplets were also presented unordered: (...) sequence order in these trials could be broken at the second or third number. A reverse distance effect of equal size was found in both anxiety groups. However, HMA participants made more judgment errors than their LMA peers when they judged one-digit counting ordered triplets. This effect was related to worse performance of HMA individuals on a symmetry span test and might be related to group differences on working memory. Importantly, HMAs were less accurate than LMA participants at rejecting unordered D2 sequences. This result is interpreted in terms of worse cognitive flexibility in HMA individuals. (shrink)

We describe an organizing framework for the study of infinitary combinatorics. This framework is Čech cohomology. It describes ZFC principles distinguishing among the ordinals of the form $\omega _n$. More precisely, this framework correlates each $\omega _n$ with an $$ -dimensional generalization of Todorcevic’s walks technique, and begins to account for that technique’s “unreasonable effectiveness” on $\omega _1$.We show in contrast that on higher cardinals $\kappa $, the existence of these principles is frequently independent of the ZFC axioms. Finally, (...) we detail implications of these phenomena for the computation of strong homology groups and higher derived limits, deriving independence results in algebraic topology and homological algebra, respectively, in the process.Abstract prepared by Jeffrey Bergfalk.E-mail: [email protected]. (shrink)

My essay positions Heidegger’s Black Notebooks (Schwarze Hefte) in the light of the later transformation of his thought after die Kehre, which introduces a new motif: “the withdrawal of Being.” And while the Jewish question disappears from his official discourse, the essay poses it nonetheless, despite and against Heidegger’s silence: Does the diagnosis from the Black Notebooks, which perceives the Jew as the agent of metaphysical destruction, still stand? In my analysis, the figurative Jew emerges in a role which Heidegger (...) refuses to recognize: as a positive agent of letting-be, acting in accordance with Being’s rhythm of self-withdrawal. (shrink)

Contemporary population ethics is dominated by views that aggregate by summing, whether of well-being or of some construct based on well-being. In contrast, average well-being is generally considered axiologically irrelevant. To many of us, however, the number of future people does not seem important, as long as it is sufficient to enable rich and varied life experiences, and as long as the population continues throughout time. It therefore seems relatively plausible to aggregate future well-being by averaging. In particular, it seems (...) plausible to value high average well-being at any particular time, and to do so for all future times. I present a time-sensitive version of the Average View that underpins such axiological intuitions. I also address a series of issues and objections that confront such a view. (shrink)

Any dynamic decision model should be based on conditional objects and must refer to (not necessarily structured) domains containing only the elements and the information of interest. We characterize binary relations, defined on an arbitrary set of conditional events, which are representable by a coherent generalized decomposable conditional measure and we study, in particular, the case of binary relations representable by a coherent conditional probability.

Mereological entities often seem to violate ‘ordinary’ ideas of what a concrete object can be like, behaving more like sets than like Aristotelian substances. However, the mereological notions of ‘part’, ‘composition’, and ‘sum’ or ‘fusion’ appear to find concrete realisation in the actual semantics of mass nouns. Quine notes that ‘any sum of parts which are water is water’; and the wine from a single barrel can be distributed around the globe without affecting its identity. Is there here, as some (...) have claimed, a ‘natural’ or ‘innocent’ form of mereology? The claim rests on the assumption that what a mass noun such as ‘wine’ denotes — the wine from a single barrel, for example — is indeed a unit of a special type, the sum or fusion of its many ‘parts’. The assumption is, however, open to question on semantic grounds. (shrink)

Milk, sand, plastic, uranium, wood, carbon, and oil are kinds of stuff. The sand in Hawaii, the uranium in North Korea, and the oil in Iraq are portions of stuff. Not everyone believes in portions of stuff.1 Those who do are likely to agree that, whatever their more specific natures, portions of stuff can at least be identified with mereological sums of their subportions.2 It seems after all trivial that a given portion of stuff just is all of its (...) subportions combined—not by a spatiotemporal or any other substantive unifying relation, but by a mere principle of summation, a principle requiring that, if some things exist, there exists the sum, or collection, of those things. (shrink)

Ordinal analysis is a research program wherein recursive ordinals are assigned to axiomatic theories. According to conventional wisdom, ordinal analysis measures the strength of theories. Yet what is the attendant notion of strength? In this paper we present abstract characterizations of ordinal analysis that address this question. -/- First, we characterize ordinal analysis as a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories, namely, the partition whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no (...) equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make. -/- Second, we characterize ordinal analysis as an ordering on arithmetically-definable and $\Pi^1_1$-sound theories, namely, the ordering wherein $T\leq U$ if the proof-theoretic ordinal of $T$ is less than or equal to the proof-theoretic ordinal of $U$. The standard ways of measuring the strength of theories are consistency strength and inclusion of $\Pi^0_1$ theorems. We introduce analogues of these notions---$\Pi^1_1$-reflection strength and inclusion of $\Pi^1_1$ theorems---in the presence of an oracle for $\Sigma^1_1$ truths, and prove that they coincide with the ordering induced by ordinal analysis. (shrink)

This paper deals with a proof theory for a theory T22 of recursively Mahlo ordinals in the form of Π2-reflecting on Π2-reflecting ordinals using a subsystem Od of the system O of ordinal diagrams in Arai 353). This paper is the first published one in which a proof-theoretic analysis à la Gentzen–Takeuti of recursively large ordinals is expounded.