Totalism is the view that one distribution of well-being is better than another just in case the one contains a greater sum of well-being than the other. Many philosophers, following Parfit, reject totalism on the grounds that it entails the repugnant conclusion: that, for any number of excellent lives, there is some number of lives that are barely worth living whose existence would be better. This paper develops a theory of welfare aggregation—the lexical-threshold view—that allows totalism to (...) avoid the repugnant conclusion, as well as its analogues involving suffering populations and the lengths of individual lives. The theory is grounded in some independently plausible views about the structure of well-being, identifies a new source of incommensurability in population ethics, and avoids some of the implausibly extreme consequences of other lexical views, without violating the intuitive separability of lives. (shrink)

This paper introduces an interesting class of predicates that come in pairs, so-called total and partial predicates. It will be shown that such predicates contribute to an explanation for the weak and strong interpretations of donkey sentences. This paper proposes that the phenomenon of weak and strong interpretations is real, and that whether a sentence receives the weak or the strong interpretation depends on the predicate in the nuclear scope of the sentence. It also proposes that sum individuals are calculated (...) at some level before the nuclear scope of the sentence is processed. Once the sum individuals are calculated, it will be decided whether the nuclear scope is true of at least one element of the sum individual (weak interpretation) or true of all elements of the sum individual (strong interpretation). (shrink)

There has been a groundswell of interest in the account of modality that Kant sets forth in his pre-Critical Only Possible Argument. Andrew Chignell's reconstruction of Kant's theistic argument in terms of what he calls has a prima facie advantage in that it appears to be able to block the plurality objection (namely, that even if every modal fact presupposes some ground, this does not entail that all modal facts share the same ground). I argue that it is both textually (...) and philosophically problematic to interpret Kant's argument in terms of real harmony. Then, I set forth an alternative response to the plurality objection which does not require the adoption of the problematic notion of real harmony. Instead, I argue that the objection can be overcome by observing that the argument seeks to ground modal facts as a totality and that, according to Kant, such relations can be accounted for only by their schematization in a single intellect. (shrink)

Armstrong famously argued in favour of introducing totality facts in our ontology. Contrary to fully negative (absence) facts, totality facts yield a theory of “moderate” or “partial” negativity, which allegedly provides an elegant solution to the truthmaking problem of negative claims and, at the same time, avoids postulating (many) first-order absences. Friends of totality facts argue that partial negativity is (i) tolerable vis-à-vis the Eleatic principle qua mark of the real, and (ii) achieves a significant advantage in terms of ontological (...) parsimony. But are totality facts, which are partially negative, really more ontologically acceptable than fully negative facts? In this paper, we argue that, comparatively, the case for totality facts is weaker than commonly assumed and that, ultimately, the answer is negative. (shrink)

In this paper, I discuss the concept of ‘shared meaning’, and the relationship between a shared understanding of signs within an animal social group and the Umwelten of individuals within the group. I explore the concept of the ‘Total Umwelt’, as described by Tønnesen, (2003), and use examples from the traditional ethology literature to demonstrate how semiotic principles can not only be applied, but underpin the observations made in animal social biology. Traditionally, neo-Darwinian theories of evolution concentrate on ‘fitness’ or (...) an organism’s capacity to survive and reproduce in its own environmental niche. However, this process also relies on underlying signs and sign processes, which are often over-looked in traditional ethology and behavioural ecology. Biosemiotics, however, places the emphasis on sign process, with signs and signals comprising a semiosphere. Significantly, whilst the semiosphere is formulated as physical phenomena, specifically energetic and material signs which can be detected and transmitted as signals from one individual to another, it is the Umwelten of living organisms which give those signals meaning. Further, two or more Umwelten can merge, giving rise to a ‘Total Umwelt’, which facilitates shared meaning of signs between two or more individuals. Across and within generations, this gives rise to cultural interpretation of signs within populations. I argue this is the fundamental basis for emergent group properties in social species, or indeed in solitary living species where individuals interact to mate, defend territories or resources, or in raising altricial young. I therefore discuss a fusion of traditional behavioural ecology- based theory with semiotics, to examine the phenomenon of ‘shared meaning’ in animal social groups. (shrink)

In Total Truth, Nancy Pearcey offers a razor-sharp analysis of the split between public and private, fact and feelings.

INTRODUCTION Ever since the beginning of the modern phenomenological movement disciplined attention has been paid to various patterns of human experience as ...

This article explores the relationship between pragmatic encroachment and epistemic permissiveness. If the suggestion that all epistemic notions are interest-relative is viable , then it seems that a certain species of epistemic permissivism must be viable as well. For, if all epistemic notions are interest relative then, sometimes, parties in paradigmatic cases of shared evidence can be maximally rational in forming competing basic doxastic attitudes towards the same proposition. However, I argue that this total pragmatic encroachment is not tenable, and, (...) thus, epistemic permissivism cannot be vindicated in this way. (shrink)

It is shown that all the provably total functions of Basic Arithmetic BA, a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which BA is sound, is introduced. This realizability is similar to Kleene's recursive realizability, except that recursive functions are restricted to primitive recursives.

Savulescu and Cameron supported selectively locking down the elderly during the COVID-19 pandemic on two grounds: first, that preserving total lockdown would entail levelling down and, second, that levelling down is wrong. Their first assumption has been thoroughly addressed, but more can be said about their wider antiegalitarian point that levelling down is simply wrong. Egalitarians are not defenceless against the levelling-down objection. Even though some consider it the most serious challenge to supporters of equality, egalitarianism possesses sound reasons to (...) assert, not only that something valuable is preserved when we level down, but also that preserving it may be, in certain circumstances, preferable to pursuing other fundamental moral goals. Although troublesome from a well-being maximising standpoint, levelling down ensures that healthcare policy reflects a commitment with the idea that people are equal in moral worth. That commitment is important enough to trump certain improvements in individual well-being. In the case of pandemic lockdowns, not all the interests protected by free movement are as fundamental as to pursue them at the cost of equality. Savulescu and Cameron’s framework is so reliant on the view that levelling down is wrong that it fails to account for the valuable loss that having the elderly suffer alone represents. (shrink)

D. Alan Shewmon has advanced a well-documented challenge to the widely accepted total brain death criterion for death of the human being. We show that Shewmon's argument against this criterion is unsound, though he does refute the standard argument for that criterion. We advance a distinct argument for the total brain death criterion and answer likely objections. Since human beings are rational animals – sentient organisms of a specific type – the loss of the radical capacity for sentience involves a (...) substantial change, the passing away of the human organism. In human beings total brain death involves the complete loss of the radical capacity for sentience, and so in human beings total brain death is death. (shrink)

A partial map f of a structure M is called almost total if |M — dom| = |M — ran| < ω. We study a difference between an almost total elementary map and an automorphism.

This paper defends a novel account of how to determine the intrinsic value of possible worlds. Section 1 argues that a highly intuitive and widely accepted account leads to undesirable consequences. Section 2 takes the first of two steps towards a novel account by clarifying and defending a view about value-contribution that is based on some of W. D. Ross’ claims about the value of pleasure. Section 3 takes the second step by clarifying and defending a view about value-suppression that (...) is based on Ross’ claims about the interplay between prima-facie duties. Section 4 states and defends the account that I call Rossian Totalism. According to this account, the atoms of intrinsic value within a world only sometimes contribute their intrinsic value to the value of that world. (shrink)

In this paper we study applicative theories of operations and numbers with the non-constructive minimum operator in the context of a total application operation. We determine the proof-theoretic strength of such theories by relating them to well-known systems like Peano Arithmetic PA and the system <0 of second order arithmetic. Essential use will be made of so-called fixed-point theories with ordinals, certain infinitary term models and Church-Rosser properties.

In this groundbreaking book David Roberts sets out to demonstrate the centrality of the total work of art to European modernism since the French Revolution.

I develop and refine an argument for the total brain death criterion of death previously advanced by Germain Grisez and me: A human being is essentially a rational animal, and so must have a radical capacity for rational operations. For rational animals, conscious sensation is a pre-requisite for rational operation. But total brain death results in the loss of the radical capacity for conscious sensation, and so also for rational operations. Hence, total brain death constitutes a substantial change—the ceasing to (...) be of the human being. Objections are considered, including the objection that total brain death need not result in the loss of capacity for sensation, and that damage to the brain less than total brain death can result in loss of capacity for rational operations. (shrink)

Wittgenstein, in the Tractatus, conceives the world as ‘the totality of facts’. Type-stratification threatens that conception : the totality of facts is an obvious example of an illegitimate totality. Wittgenstein’s notion of truthoperation evidently has some role to play in avoiding that threat, allowing propositions, and so facts, to constitute a single type. The paper seeks to explain that role in a way that integrates the ‘philosophical’ and ‘technical’ pressures on the notion of an operation.

Total work of art in an age of mechanical reproduction -- Total stage: Wagner's festspielhaus -- Total machine: the Bauhaus theatre -- Total montage: Brecht's reply to Wagner -- Total state: Riefenstahl's triumph of the will -- Total world: Disney's theme parks -- Total vacuum: Warhol's performances -- Total immersion: cyberspace.

Essays by 14 Levinas scholars provide a fresh acount of the argument and purpose of Emmanuel Levinas's major work, Totality and Infinity, drawing parallels between Levinas and other thinkers; considering Levinas's relationship to other disciplines such as nursing, psychotherapy, and law; and bringing this seminal text to bear on specific, concrete issues of present-day concern"--Provided by publisher.

The image of the hospital is presented to the public as a place of healing. Though the oft‐criticized total institutions of the past have been notably dismantled, the totalizing practices therein are now operationalized in the health care system. Through the lens of Erving Goffman, this article offers ways in which health care institutions operationalize totalizing practices, contributing to the mortification of patients and nurses alike in service to the bureaucratic machine. This article examines the ways in which totalizing practices (...) may disrupt the agency of both patients and nurses alike. (shrink)

Taking its clues from Popperian philosophy of science, cladistics adopted a number of assumptions of the empiricist tradition. These include the identification of a dichotomy between observation reports and theoretical statements and its subsequent abandonment on the basis of the insight that all observation reports are theory-laden. The neglect of the ‘context of discovery’, which is the step of theory (hypothesis) generation. The emphasis on coherentism in the ‘context of justification’, which is the step of evaluation of the relative merits (...) of alternative theories. The appeal to a total evidence approach in phylogenetic inference. And finally, a silence about causation, which results in an instrumentalist approach to phylogeny reconstruction. This paper explores how these empiricist assumptions are embedded in phylogenetic systematics, and why these assumptions are problematic for cladists (or any taxonomists). (shrink)

Armstrong appeals to the existence of totalities in order to solve the problem of negative truths. The totality of first-order states of affairs is a truthmaker for all negative truths, but it involves things which are irrelevant to many such truths. To solve this problem, Armstrong claimed that negative truths have minimal truthmakers which usually consist in totalities smaller than the totality of first-order states of affairs. Merricks objects to this claim by arguing that given Armstrong’s definition of minimal truthmakers, (...) the totality of first-order states of affairs is a minimal truthmaker for every negative truth. In this article, I respond to Merricks’s objection. I will first show that the definition of minimal truthmakers that he proposes is not plausible. However, Merricks’s objection, I will argue, also works on the standard definition of minimal truthmakers. I will then show that for independent reasons, the standard definition should be revised, and, given my revised definition, Merricks’s objection fails. (shrink)

Dummctt argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept, since in such cases the objects are not the required definite totality. The chief problem in understanding this complex argument is the crucial but unexplained phrase 'definite totality' and the associated claim that it follows from the intuitive notion of set that the objects over which a classical quantifier ranges form a set. 'Definite totality' is best understood as (...) disguised plural talk like Cantor's 'consistent multiplicity', although this does not help in understanding how a totality could be anything other than definite. Moreover, contrary to his claims, Dummett's own notion of set is not intuitive and he does not demystify the set-theoretic paradoxes. In conclusion, it is argued that Dummett's context principle is responsible for the incoherent projection of the haziness of a conception of some objects onto reality. (shrink)

In this essay I argue that Kant remained committed to the necessity and fundamental importance of education throughout his career. Like Johann Bernhard Basedow (1724–90), Kant holds that a ‘total transformation’ of schools is necessary, and he holds this view not only in the 1770s but in his later years as well. In building my case I try to refute two recent opposing interpretations – Reinhard Brandt’s position that Kant’s early ‘education enthusiasm’ was later replaced by a politics enthusiasm, and (...) Manfred Kuehn’s view that the increasing importance of autonomy in Kant’s mature ethics leads him to de-emphasize education. (shrink)

Building upon his previous books about Marx, Hayek, and Rand, _Total Freedom_ completes what _Lingua Franca_ has called Sciabarra’s "epic scholarly quest" to reclaim dialectics, usually associated with the Marxian left, as a methodology that can revivify libertarian thought. Part One surveys the history of dialectics from the ancient Greeks through the Austrian school of economics. Part Two investigates in detail the work of Murray Rothbard as a leading modern libertarian, in whose thought Sciabarra finds both dialectical and nondialectical elements. (...) Ultimately, Sciabarra aims for a dialectical-libertarian synthesis, highlighting the need to think of the "totality" of interconnections in a dynamic system as the way to ensure human freedom while avoiding "totalitarianism". (shrink)

Dummctt argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept, since in such cases the objects are not the required definite totality. The chief problem in understanding this complex argument is the crucial but unexplained phrase 'definite totality' and the associated claim that it follows from the intuitive notion of set that the objects over which a classical quantifier ranges form a set. 'Definite totality' is best understood as (...) disguised plural talk like Cantor's 'consistent multiplicity', although this does not help in understanding how a totality could be anything other than definite. Moreover, contrary to his claims, Dummett's own notion of set is not intuitive and he does not demystify the set-theoretic paradoxes. In conclusion, it is argued that Dummett's context principle is responsible for the incoherent projection of the haziness of a conception of some objects onto reality. (shrink)

Berger, U., Total sets and objects in domain theory, Annals of Pure and Applied Logic 60 91-117. Total sets and objects generalizing total functions are introduced into the theory of effective domains of Scott and Ersov. Using these notions Kreisel's Density Theorem and the Theorem of Kreisel-Lacombe-Shoenfield are generalized. As an immediate consequence we obtain the well-known continuity of computable functions on the constructive reals as well as a domain-theoretic characterization of the Heriditarily Effective Operations.

In Preference, Value, Choice, and Welfare, I argued, among other things, that preferences in economics are and ought to be total subjective comparative evaluations, that the theory of rational choice is a reformulation of everyday folk-psychological explanations and predictions of behaviour, and that revealed preference theory is completely untenable. All three of these theses have been challenged in essays by Erik Angner (2018), Francesco Guala (2019) and Johanna Thoma (2021a, 2021b). This essay responds to these criticisms and defends these three (...) theses. (shrink)

In his masterpiece The Star of Redemption, Franz Rosenzweig shows as the notion of totality is a constant and central reference in the history of philosophy from Ionia to Jena. This paper aims to explain a different meaning of the concept of totality, reconsidering some aspects of the question starting from the philosophical reflection of Franz Rosenzweig and his opposition to the Hegelian thought. In particular, according to Rosenzweig, the concept of totality is the essential background in which one could (...) rethink the concept of community. For this reason, the second part of this paper is focused on the implications of a different concept of totality embodied by Judaism. (shrink)

This article investigates the connection and dependence between the definiteness of the totalities involved in mathematical structures and the determinateness of statements about that structure. From a logical perspective, we investigate whether logical principles expressing the definiteness of totalities license the use of classical logic. From a philosophical perspective, this article provides a reconstruction of Solomon Feferman’s claim that the definiteness of the natural number conception implies the determinateness of arithmetical statements and therefore justifies the adoption of classical logic for (...) arithmetical theories. (shrink)

Coherence-spaces and domains with totality are used to give interpretations of inductively defined types. A category of coherence spaces with totality is defined and the closure of positive inductive type constructors is analysed within this category. Type streams are introduced as a generalisation of types defined by strictly positive inductive definition. A semantical analysis of type streams with continuous recursion theorems is established. A hierarchy of domains with totality defined by positive induction is defined, and density for a sub-hierarchy is (...) proved. (shrink)

When in 2001 Earth Liberation Front activists drove metal spikes into hundreds of trees in Gifford Pinchot National Forest, they were protesting the sale of a section of the old-growth forest to a timber company. But ELF's communiqu on the action went beyond the radical group's customary brief. Drawing connections between the harms facing the myriad animals who make their home in the trees and the struggles for social justice among ordinary human beings resisting exclusion and marginalization, the dispatch declared, (...) "all oppression is linked, just as we are all linked," and decried the "patriarchal nightmare" in the form of "techno-industrial global capitalism." In Total Liberation, David Naguib Pellow takes up this claim and makes sense of the often tense and violent relationships among humans, ecosystems, and nonhuman animal species, expanding our understanding of inequality and activists' uncompromising efforts to oppose it. Grounded in interviews with more than one hundred activists, on-the-spot fieldwork, and analyses of thousands of pages of documents, websites, journals, and zines, Total Liberation reveals the ways in which radical environmental and animal rights movements challenge inequity through a vision they call "total liberation." In its encounters with such infamous activists as scott crow, Tre Arrow, Lauren Regan, Rod Coronado, and Gina Lynn, the book offers a close-up, insider's view of one of the most important--and feared--social movements of our day. At the same time, it shows how and why the U.S. justice system plays to that fear, applying to these movements measures generally reserved for "jihadists"--with disturbing implications for civil liberties and constitutional freedom. How do the adherents of "total liberation" fight oppression and seek justice for humans, nonhumans, and ecosystems alike? And how is this pursuit shaped by the politics of anarchism and anticapitalism? In his answers, Pellow provides crucial in-depth insight into the origins and social significance of the earth and animal liberation movements and their increasingly common and compelling critique of inequality as a threat to life and a dream of a future characterized by social and ecological justice for all. (shrink)

We consider the classical concept of time of permanence and observe that its quantum equivalent is described by a bona fide self-adjoint operator. Its interpretation, by means of the spectral theorem, reveals that we have to abandon not only the idea that quantum entities would be characterizable in terms of spatial trajectories but, more generally, that they would possess the very attribute of spatiality. Consequently, a permanence time shouldn’t be interpreted as a “time” in quantum mechanics, but as a measure (...) of the total availability of a quantum entity in participating to a process of creation of a spatial localization. (shrink)

Israel Scheffler's ground-breaking essay, On Human Potential, deserves to be more widely known among educational policy analysts, especially in light of the popularity in educationist circles of W.E. Deming's organizational philosophy known as Total Quality Management . In what follows,I argue that the heuristical value of Deming's perscriptions are entailed in Scheffler's On Human Potential. More importantly, I argue, where Deming's work falls short, especially in being naive about the human condition, Scheffler's analysis provides a foundation for management theory in (...) education that insures the flourishing of an optimal number of contributing participants. In fashioning these ideas, Scheffler brings pioneering thinking to the emerging fields of management studies generally and educational policy specifically. (shrink)

The complete system of knowledge is a standard trope of science fiction, a techno-utopian dream and an aesthetic ideal. It is Solomon’s House, the Encyclopaedia and the Museum. It is also an ideology – of Enlightenment, High Modernism and absolute governance. Far from ending the dream of a total archive, 20th-century positivist rationality brought it ever closer. From Paul Otlet’s ‘Mundaneum’ to Mass-Observation, from the Unity of Science movement to Wikipedia, the dream of universal knowledge dies hard. As a political (...) tool, the total archive encompasses population statistics, gross domestic product, indices of the Standard of Living and the international ideology of the United Nations Educational, Scientific and Cultural Organization, the World Health Organization, the free market and, most recently, Big Data. Questions of the total archive engage key issues in the philosophy of classification, the poetics of the universal, the ideology of surveillance and the technologies of information retrieval. What are the social structures and political dynamics required to sustain total archives, and what are the temporalities implied by such projects? This introduction and the articles that follow describe and place in historical context a series of concrete instances of totality. Our analysis is arranged according to four central themes: the relationship between the Archive (singular) and archives (plural); the image of the archive and the aesthetics of totality; pathologies of accumulation; and the specific historical trajectory of the total archive in the 19th and 20th centuries. (shrink)

Let us begin—as is fitting—with what is familiar. Now, familiar is by no means what ‘totalizing thinking’ is, but rather what ‘totalitarian’ means. Today this word rings in everyone’s ears, is common in public life—not so much as a descriptive, but more as a defamatory epithet. It brands political systems of our time which not only gave rise to horrifying consequences, but what is more spread terror through their form of domination. Now, rule by means of terror is in fact (...) no recent discovery; new here is rather the technique of the complete penetration of every expression of social life. Were the latter perhaps susceptible in a special way to such penetration? (shrink)

We tackle the problem of preservation of totality by composition in arena games. We first explain how this problem reduces to a finiteness theorem on what we call pointer structures, similar to the parity pointer functions of Harmer, Hyland and Mélliès and the interaction sequences of Coquand. We discuss how this theorem relates to normalization of linear head reduction in simply-typed lambda-calculus, leading us to a semantic realizability proof à la Kleene of our theorem. We then present another proof of (...) a more combinatorial nature. Finally, we discuss the exact class of strategies to which our theorems apply. (shrink)

Hand-lettered inspirational quotes paired with contemporary illustrations for more than a year of Motivational Mondays, You’ve Totally Got This! has got you covered. Sometimes all we need is a little nudge: this book serves as inspiration for those facing new beginnings, a sudden change, or just a whole lot to do. Great for grads, creatives, and entrepreneurs: every spread is like a greeting card, so this delightful volume serves as a stand-alone championing or as the perfect finishing touch to a (...) gift. (shrink)

In The Principles of Art, R. G. Collingwood pursues, on the one hand, a ‘definition’ of art, and, on the other, a ‘metaphysics’. The Principles is divided into three Books. Book I is devoted mostly to craft, while Book II pertains largely to metaphysics. The fact that Book II is twice the size of Book III, where the discussion of ‘art proper’ takes place, is proof enough that the metaphysical part of the Principles is not a mere excursus. Collingwood’s ontology (...) is indispensable for understanding his aesthetics, and vice versa. The crucial link is the imagination. What Collingwood calls ‘total imaginative experience’ is described in the Principles as the sine qua non of both thought and sensibility. The aim of this article is to examine the ontological import of Collingwood’s conception of the total imagination. (shrink)

Units, rules, codes, systems: this is how most linguists study language. Integrationalists such as Michael Toolan, however, focus instead on how language functions in seamless tandem with the rest of human activity. In Total Speech, Toolan provides a clear and comprehensive account of integrationalism, a major new theory of language that declines to accept that text and context, language and world, are distinct and stable categories. At the same time, Toolan extends the integrationalist argument and calls for a radical change (...) in contemporary theorizing about language and communication. In every foundational area of linguistics--from literal meaning and metaphor to the nature of repetition to the status of linguistic rules--Toolan advances fascinating and provocative criticisms of received linguistic assumptions. Drawing inspiration from the writings of language theorist Roy Harris, Toolan brings the integrationalist perspective to bear on legal cases, the reception of Salman Rushdie, poetry, and the language of children. Toolan demonstrates that the embeddedness of language and the situation-sensitive mutability of meaning reveal language as a tool for re-fashioning and renewal. Total Speech breaks free of standard linguistics' fascinated attraction with "cognitive blueprints" and quasi-algorithmic processing to characterize language anew. Toolan's reflections on the essence of language, including his important discussion of intention, have strong implications for students and scholars of discourse analysis, literature, the law, anthropology, philosophy of language, communication theory, and cognitive science, as well as linguistics. (shrink)

Pragmatic encroachment in epistemology is the idea that whether one knows some proposition depends on whether one can rely on it practically. Total pragmatic encroachment affirms that practical considerations of this sort encroach not just on knowledge but on all interesting normative epistemic statuses a belief might have. Ichikawa, Jarvis, and Rubin (2012) have argued that this stronger thesis conflicts with mainstream belief‐desire psychology. Worse still, they argue that attempting to defend the thesis gets one caught in vicious circularities. The (...) aim of this paper is to show that, if we are careful in how we understand the key idea of being sensitive to practical considerations, we can defend total pragmatic encroachment and avoid the circularities. In fact, depending on how it is understood, we can even square mainstream belief‐desire psychology with total pragmatic encroachment as well. (shrink)

Let be a countable first‐order language and be an ‐structure. “Definable set” means a subset of M which is ‐definable in with parameters. A set is said to be immune if it is infinite and does not contain any infinite definable subset. X is said to be partially immune if for some definable A, is immune. X is said to be totally non‐immune if for every definable A, and are not immune. Clearly every definable set is totally non‐immune. Here we (...) ask whether the converse is true and prove that it is false for every countable structure whose class of definable sets satisfies a mild condition. We investigate further the possibility of an alternative construction of totally non‐immune non‐definable sets with the help of a subclass of immune sets, the class of cohesive sets, as well as with the help of a generalization of definable sets, the semi‐definable ones (the latter being naturally defined in models of arithmetic). Finally connections are found between totally non‐immune sets and generic classes in nonstandard models of arithmetic. (shrink)

We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns (...) out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve the solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model. (shrink)

We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns (...) out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve the solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model. (shrink)