In an ideal democracy, people from all walks of life would come together to talk meaningfully and respectfully about politics. But we do not live in an ideal democracy. In contemporary democracies, which are marked by deep social divisions, different groups for the most part avoid talking to each other. And when they do talk to each other, their speech often seems to be little more than a vehicle for rage, hatred, and deception. -/- Democratic Speech in Divided Times argues (...) that we should nevertheless not give up on the ideal of democratic public speech. Drawing on the resources of political theory, epistemology, and philosophy of language, this book develops a sustained account of the norms that should govern public discourse in deeply divided circumstances. Should we try to find common ground when we talk to our political opponents, even though they seem unreasonable? Should we refrain from expressing anger, if we want to get things done? How can we use our speech to fight hate speech and disinformation? And is it even possible to speak to 'the other side,' in settings where different groups dislike one another, live apart from one another, and don't know much about one another? By tackling these questions, Lepoutre demonstrates that, when governed by the right set of norms, public speech can be a powerful force for good even amidst profound social divisions. (shrink)

Human activity is permeated by norms of all sorts: moral norms provide the 'code' for what we ought to do, norms of logic regulate how we ought to reason, scientific norms set the standards for what counts as knowledge, legal norms determine what is lawfully permitted and what isn't, aesthetic norms establish canons of beauty and shape artistic trends and practices, and socio-cultural norms provide criteria for what counts as tolerable, just, praiseworthy, or unacceptable in a community or milieu. Given (...) the diversity of norm-governed phenomena prevailing in our everyday experience, it is not surprising that the question of normativity has recently generated important debates in philosophy. However, the more specific question concerning the nature and function of 'norms in perceptual experience' has received comparatively little attention. This volume brings together scholars from philosophy of mind, philosophy of perception, and phenomenology to explore this question. (shrink)

Men are living in the midst of a world pervaded by ideas.* Ideas, which provide men with help in their will to act and to think according to rules, which offer them guidance in their lives, are mustered into systems which are called ideologies.Ideologies are issued by the inclusive society, by special, functional groups inside the society and, beginning with a given stage in the development of human society, by ideological movements which profess to provide men not only with guidance (...) for special tasks or functions, but for life as a whole. So, individual men are faced with various sets of ideologies, more or less in agreement between themselves, more 'or less authoritative for them. They must choose between the suggestions, the rules, the orders given to them. As men are always in search of some degree of consistency in their ways of thinking and acting, they fit together the ideas they choose, unconsciously, into systems, into personal, private, individual ideologies. (shrink)

This is a survey paper giving a self-contained account of Shelah's theory of the pcf function pcf={cf:D is an ultrafilter on a}, where a is a set of regular cardinals such that a

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Traditional approaches describe ethical decision-making of leaders as driven by conscious deliberation and analysis. Accordingly, existing approaches of ethically-oriented leader development usually focus on the promotion of deliberative ethical decision-making, based on normative knowledge and moral reasoning. Yet, a continually growing body of research indicates that a considerable part of moral functions involved in ethical decision-making is automatic and intuitive. In this article, we discuss the implications of this moral intuition approach for the domain of ethically-oriented leader development. Specifically, we (...) introduce a conceptual model and develop a set of theoretical propositions, suggesting that the moral intuition perspective significantly contributes to effective ethically-oriented leader development. The discussion examines theoretical implications and practical applications of the presented propositions and outlines directions for future research. (shrink)

Faced with hate speech, oppressed groups can use their own speech to respond to their verbal oppressors. This “counterspeech,” however, sometimes itself takes on a hateful form. This paper explores the moral standing of such “hateful counterspeech.” Is there a fundamental moral asymmetry between hateful counterspeech, and the hateful utterances of dominant or oppressive groups? Or are claims that such an asymmetry exists indefensible? I argue for an intermediate position. There _is_ a key moral asymmetry between these two forms of (...) speech. But, this asymmetry notwithstanding, hateful counterspeech is capable of enacting serious harms—and so, contrary to what many legal theorists have argued, it is in principle an appropriate object of legal regulation. I begin by considering the central argument for thinking that hateful counterspeech is not seriously troubling. This argument holds that oppressed groups lack authority—and, by extension, “speaker power.” Yet this argument, I suggest, sits in tension with the fact that low-status members of dominant groups can, through their utterances, seriously harm members of oppressed groups. Philosophers of language have developed sophisticated arguments to explain this last phenomenon: they have argued that speaker power is relativised to particular jurisdictions; that it can be acquired dynamically in local settings; and that it is socially dispersed. I argue that, in light of these arguments, it appears that hateful counterspeech, too, can generate serious harms. Nevertheless, I show that this conclusion is compatible with recognising a crucial moral asymmetry between hateful counterspeech and the hate speech of oppressors. (shrink)

Intensity profiles of emotional experience over time have been found to differ primarily in explosiveness and accumulation. However, the determinants of these temporal features remain poorly understood. In two studies, we examined whether emotion regulation strategies are predictive of the degree of explosiveness and accumulation of negative emotional episodes. Participants were asked to draw profiles reflecting changes in the intensity of emotions elicited either by negative social feedback in the lab or by negative events in daily life. In addition, trait, (...) and state usage of a set of emotion regulation strategies was assessed. Multilevel analyses revealed that trait rumination was positively associated with emotion accumulation. State rumination was also positively associated with emotion accumulation and, to a lesser extent, with emotion explosiveness. These results provide support for emotion regulation theories, which hypothesise that rumination is a central mechanism underlying the maintenance of negative emotions. (shrink)

We investigate the cofinality of the partial order N κ of functions from a regular cardinal κ into the ideal N of Lebesgue measure zero subsets of R. We show that when add(N) = κ and the covering lemma holds with respect to an inner model of GCH, then cf(N κ ) = max {cf(κ κ ), cf([ cf(N)] κ )}. We also give an example to show that the covering assumption cannot be removed.

Summary The article deals with a complex of issues connected with the campaign waged by the Macedonian expeditionary corps in Asia Minor in 336–335 BC. The author clears up the aims set for the advance-guard, its command structure, strength and composition. He also describes the relevant military operations and reveals the reasons both for the Macedonians’ successes in 336 and their failures in 335. The idea is argued that despite the final failures, it is hardly possible to say that the (...) campaign the expeditionary corps conducted ended in its total defeat. Besides, it is noted that those military operations had major significance for Alexander’s campaign in Asia Minor in 334, because a number of preconditions for its full success had been created right in their course. (shrink)

The claim that language is in principle inaccessible to consciousness may look counterintuitive but is not as challenging as finding an answer to the subsequent question of why that must be the case -- if language is a function that is in the service of consciousness and we cannot imagine why language would have existed at all without the existence of consciousness. On the one hand, language is the cognitive capacity that seems best fit to support consciousness in its monitoring (...) and control functions; on the other hand, language learning (learning the rules of one's own language), language structure and language processing turn out upon closer scrutiny to be in principle inaccessible to consciousness. I present a set of arguments in favour of the thesis that language is in principle inaccessible to consciousness on the basis of a set of asymmetries between sentence structure and the structure of consciousness. If the thesis in question is on the right track, we have to face two basic problems. The first deals with linguistics method(s), namely how can we study a very complex mental phenomenon like language if it is not available to introspection? The second problem is related to the question put in the title of this article. The suggested answer is along the lines that inaccessibility of language to consciousness enables a cognitive architecture that can run a Cartesian theatre. (shrink)

The recent publication of Geschlecht III allows us, for the first time, to synoptically embrace the Geschlecht project and to appreciate its overarching consistency. I choose to read the Geschlecht series as the scene of an inheritance whereby Derrida wrestles with Heidegger’s legacy. More specifically, I identify the linchpin of Derrida’s critique of gathering (Versammlung), which organizes his reading throughout the series. I then distinguish between two layers within this reading : the first, wherein he stages the refusal of an (...) inheritance deemed metaphysical or suspicious, and the less well-known second, which sets the stage for a shared heritage (gift, promise, economy). The value of gathering thus consecutively appears on one and the other side of these dividing lines, until the legacy’s pluralization turns out to be what’s at stake in the scene. (shrink)

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing. The corresponding theorem for the meager ideal was (...) established by Bartoszyński and Kada. (shrink)

We give a proof of Hechler's theorem that any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\aleph_1$\end{document}-directed partial order can be embedded via a ccc forcing notion cofinally into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\omega^\omega$\end{document} ordered by eventual dominance. The proof relies on the standard forcing relation rather than the variant introduced by Hechler.

In this paper we prove that any Δ30 degree has an increasing η -representation. Therefore, there is an increasing η -representable set without a strong η -representation.

A great deal has been made of the question of whether nano-materials provide a unique set of ethical challenges. Equally important is the question of whether they provide a unique set of regulatory challenges. In the last 18 months, the US Environmental Protection Agency has begun the process of trying to meet the regulatory challenge of nano using the Toxic Substances Control Act (1976)(TSCA). In this central piece of legislation, ‘newness’ is a critical concept. Current EPA policy, we argue, does (...) not adequately (or ethically) deal with the novelty of nano. This paper is an exploration of how to do a better job of accounting for nanomaterials as ‘new.’ We explore three alternative ways that nanomaterials might be made to fall under the TSCA regulatory umbrella. Since nanomaterials are of interest precisely because of the exciting new properties that emerge at the nano-scale, each of these three alternatives must meet what we call the ‘novelty condition’ and avoid what we call the ‘central paradox’ of existing regulatory policy. Failure to meet either of these conditions is a moral failure. We examine both the strengths and weaknesses of each alternative in order to illuminate the conceptual, practical, and moral challenges of novelty. (shrink)

A great deal has been made of the question of whether nano-materials provide a unique set of ethical challenges. Equally important is the question of whether they provide a unique set of regulatory challenges. In the last 18 months, the US Environmental Protection Agency has begun the process of trying to meet the regulatory challenge of nano using the Toxic Substances Control Act (1976)(TSCA). In this central piece of legislation, ‘newness’ is a critical concept. Current EPA policy, we argue, does (...) not adequately (or ethically) deal with the novelty of nano. This paper is an exploration of how to do a better job of accounting for nanomaterials as ‘new.’ We explore three alternative ways that nanomaterials might be made to fall under the TSCA regulatory umbrella. Since nanomaterials are of interest precisely because of the exciting new properties that emerge at the nano-scale, each of these three alternatives must meet what we call the ‘novelty condition’ and avoid what we call the ‘central paradox’ of existing regulatory policy. Failure to meet either of these conditions is a moral failure. We examine both the strengths and weaknesses of each alternative in order to illuminate the conceptual, practical, and moral challenges of novelty. (shrink)

In the history of science, the birth of classical chemistry and thermodynamics produced an anomaly within Newtonian mechanical paradigm: force and acceleration were no longer citizens of new cited sciences. Scholars tried to reintroduce them within mechanistic approaches, as the case of the kinetic gas theory. Nevertheless, Thermodynamics, in general, and its Second Law, in particular, gradually affirmed their role of dominant not-reducible cognitive paradigms for various scientific disciplines: more than twenty formulations of Second Law—a sort of indisputable intellectual wealth—are (...) conceived after 1824 Sadi Carnot’s original statement and a multitude of entropy functions are proposed after 1865 Clausius’ former definition. Furthermore, at the end of nineteenth century, thermodynamics extended its cognitive domain to chemistry. Mainly thanks to Gibbs, a brand new discipline—chemical thermodynamics or physical chemistry—gradually affirmed its role inside the scientific community. This paper reports the former results of collaborative research program in the History and Epistemology of Science as well as Nature of Science Teaching aimed at retracing the foundations of the physical chemistry. Specifically, the research is structured in three parts: historical-epistemic reflections on fundamental thermodynamic concepts and principles—such as reversible process, heat, temperature, thermal equilibrium and Clausius’ Second Law—that play a structural role inside modern physical chemistry; panoramic overview on the entropy, whose polysemy makes it one of the most demanding concepts for scholars, teachers and students while approaching thermodynamics; conceptualization of chemical equilibrium as complex entity according to the dual epistemological approach offered by Gibbs’ thermodynamic model and the kinetic standpoint by Guldberg and Waage. In particular, the present work details an original reading of thermodynamic principles with the aim of setting forth a rationalized multidisciplinary substrate whereon the foundational concepts of reversible process and thermal equilibrium can be set. (shrink)

In the history of science, the birth of classical chemistry and thermodynamics produced an anomaly within Newtonian mechanical paradigm: force and acceleration were no longer citizens of new cited sciences. Scholars tried to reintroduce them within mechanistic approaches, as the case of the kinetic gas theory. Nevertheless, Thermodynamics, in general, and its Second Law, in particular, gradually affirmed their role of dominant not-reducible cognitive paradigms for various scientific disciplines: more than twenty formulations of Second Law—a sort of indisputable intellectual wealth—are (...) conceived after 1824 Sadi Carnot’s original statement and a multitude of entropy functions are proposed after 1865 Clausius’ former definition. Furthermore, at the end of nineteenth century, thermodynamics extended its cognitive domain to chemistry. Mainly thanks to Gibbs, a brand new discipline—chemical thermodynamics or physical chemistry—gradually affirmed its role inside the scientific community. This paper reports the former results of collaborative research program in the History and Epistemology of Science as well as Nature of Science Teaching aimed at retracing the foundations of the physical chemistry. Specifically, the research is structured in three parts: historical-epistemic reflections on fundamental thermodynamic concepts and principles—such as reversible process, heat, temperature, thermal equilibrium and Clausius’ Second Law—that play a structural role inside modern physical chemistry; panoramic overview on the entropy, whose polysemy makes it one of the most demanding concepts for scholars, teachers and students while approaching thermodynamics; conceptualization of chemical equilibrium as complex entity according to the dual epistemological approach offered by Gibbs’ thermodynamic model and the kinetic standpoint by Guldberg and Waage. In particular, the present work details an original reading of thermodynamic principles with the aim of setting forth a rationalized multidisciplinary substrate whereon the foundational concepts of reversible process and thermal equilibrium can be set. (shrink)

We show that if M is an r‐maximal set, A is a major subset of M, B is an arbitrary set and, then. We prove that the c.e. ‐degrees are not dense. We also show that there exist infinite collections of ‐degrees and such that the following hold: (i) for every i, j,, and,(ii) each consists entirely of r‐maximal sets, and(iii) each consists entirely of non‐r‐maximal hyperhypersimple sets.

Soare [20] proved that the maximal sets form an orbit in${\cal E}$. We consider here${\cal D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of${\cal D}$-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the${\cal D}$-maximal sets. Although these invariants help us (...) to better understand the${\cal D}$-maximal sets, we use them to show that several classes of${\cal D}$-maximal sets break into infinitely many orbits. (shrink)

We show that the c.e. Q1,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q_{1,N}}$$\end{document}-degrees are not an upper semilattice. We prove that if M is an r-maximal set, A is an arbitrary set and M≡Q1,NA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M \equiv{}_ {Q_{1,N}}A}$$\end{document}, then M≤mA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M\leq{}_{m} A}$$\end{document}. Also, if M1 and M2 are r-maximal sets, A and B are major subsets of M1 and M2, respectively, and (...) M1\A≡Q1,NM2\B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_{1}{\setminus} A\equiv{}_{Q_{1,N}}M_{2}{\setminus} B}$$\end{document}, then M1\A≡mM2\B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_{1}{\setminus}A\equiv{}_{m}M_{2}{\setminus} B}$$\end{document}. If M1 and M2 are r-maximal sets and M10,M11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_{1}^{0},\,M_{1}^{1}}$$\end{document} and M20,M21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_{2}^{0},\,M_{2}^{1}}$$\end{document} are nontrivial splittings of M1 and M2, respectively, then M10≡Q1,NM20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_{1}^{0} \equiv{}_{Q_{1,N}}M_{2}^{0}}$$\end{document} if and only if M10≡1M20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_{1}^{0} \equiv{}_{1}M_{2}^{0}}$$\end{document}. From this result follows that if A and B are Friedberg splitting of an r-maximal set, then the Q1,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q_{1,N}}$$\end{document}-degree of A contains only one c.e. 1-degree. (shrink)

Soare [20] proved that the maximal sets form an orbit in${\cal E}$. We consider here${\cal D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of${\cal D}$-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the${\cal D}$-maximal sets. Although these invariants help us (...) to better understand the${\cal D}$-maximal sets, we use them to show that several classes of${\cal D}$-maximal sets break into infinitely many orbits. (shrink)

We prove that if V = L then there is a $\Pi _{1}^{1}$ maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.

We study the filter ℒ*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ*(A) is still Σ0 3-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ*(A).

The main results of the present paper are the following theorems: 1. There is no e ∈ ω such that for any A, B ⊆ ω, SA = Wmath image is simple in A, and if A′ [TRIPLE BOND]TB′, then SA =* SB. 2 There is an e ∈ ω such that for any A, B ⊆ ω, MA = We is incomplete maximal in A, and if A =* B, then MA [TRIPLE BOND]TMB.

The structure of the l-degrees included in an m-degree with a maximal set together with the l-reducibility relation is characterized. For this a special sublattice of the lattice of recursively enumerable sets under the set-inclusion is used.

Martin [4, Theorems 1 and 2] proved that a Turing degree a is the degree of a maximal set if, and only if, a′ = 0″. Lachlan has shown that maximal sets have minimal many-one degrees [2, §1] and that every nonrecursive r.e. Turing degree contains a minimal many-one degree [2, Theorem 4]. Our aim here is to show that any r.e. Turing degree a of a maximal set contains an infinite number of maximal sets whose (...) many-one degrees are pairwise incomparable; hence such Turing degrees contain an infinite number of distinct minimal many-one degrees. This theorem has been proved by Yates [6, Theorem 5] in the case when a = 0′. The need for this theorem first came to our attention as a result of work done by the author [3, Theorem 2.3]. There we looked at the structure / obtained from the recursive functions of one variable under the equivalence relation f ~ g if, and only if, f(x) = g(x) a.e., that is, for all but finitely many x ∈, where M is a maximal set, and M is its complement. We showed that / 1 ≡ / 2 if, and only if, 1 = m 2, i.e., 1. and 2 have the same many-one degree. However, it might be possible that some Turing degree of a maximal set contains exactly one many-one degree of a maximal set. Theorem 1 was proved to show that this was not the case, and hence that the theory of / is not an invariant of Turing degree. (shrink)

Related Works: Part II: C. T. Chong, Yue Yang. $\Sigma_2$ Induction and Infinite Injury Priority Argument, Part II: Tame $\Sigma_2$ Coding and the Jump Operator. Ann. Pure Appl. Logic, vol. 87, no. 2, 103--116. Mathematical Reviews : MR1490049 Part III: C. T. Chong, Lei Qian, Theodore A. Slaman, Yue Yang. $\Sigma_2$ Induction and Infinite Injury Priority Argument, Part III: Prompt Sets, Minimal Paries and Shoenfield's Conjecture. Mathematical Reviews : MR1818378.

In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) of maximality principles. (shrink)

The paper generalizes Van McGee's well-known result that there are many maximal consistent sets of instances of Tarski's schema to a number of non-classical theories of truth. It is shown that if a non-classical theory rejects some classically valid principle in order to avoid the truth-theoretic paradoxes, then there will be many maximal non-trivial sets of instances of that principle that the non-classical theorist could in principle endorse. On the basis of this it is argued that the idea (...) of classical recapture, which plays such an important role for non-classical logicians, can only be pushed so far. (shrink)

Paul Horwich (1990) once suggested restricting the T-Schema to the maximally consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set (...) and should be restricted as little as possible. The view might even have been held by Ernst Zermelo (1908), who,according to Penelope Maddy (1988), subscribed to a ‘one step back from disaster’ rule of thumb: if a natural principle leads to contra-diction, the principle should be weakened just enough to block the contradiction. We prove a generalization of McGee’s Theorem, anduse it to show that the situation for set theory is the same as that for truth: there are multiple incompatible sets of instances of Naïve Comprehension, none of which, given minimal assumptions, is recursively axiomatizable. This shows that the view adumbrated by Goldstein, Quine and perhaps Zermelo is untenable. (shrink)

In the maximal form of indirect development found in many taxa of marine invertebrates, embryonic cell lineages of fixed fate and limited division capacity give rise to the larval structures. The adult arises from set‐aside cells in the larva that are held out from the early embryonic specification processes, and that retain extensive proliferative capacity. We review the locations and fates of set‐aside cells in two protostomes, a lophophorate and a deuterostome. The distinct adult body plans of many phyla (...) develop from homologous set‐aside cells within homologous larvae. We argue that the stocks from which these phyla arose utilized these respective larvae, and the diversity of their adult body plans reflects diverse pattern formation processes executed in their set‐aside cell populations. Chordates and arthropods develop directly, but share adult characters with indirectly developing phyla. Thus the deuterostome and protostome stocks that were ancestral to chordates and arthropods, respectively, also utilized maximal indirect development. (shrink)

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, (...) and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account. (shrink)

A characterization of groups definable in o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role of maximal tori is played by maximal 0-subgroups. Along the way we give structural theorems for solvable groups, linear groups, and extensions of definably compact by torsion-free definable groups.

In a recent paper, while discussing the role of the notion of analyticity in Carnap’s thought, Howard Stein wrote: “The primitive view–surely that of Kant–was that whatever is trivial is obvious. We know that this is wrong; and I would put it that the nature of mathematical knowledge appears more deeply mysterious today than it ever did in earlier centuries – that one of the advances we have made in philosophy has been to come to an understanding of just ∗I (...) am grateful to audiences at the Steinfest, University of Chicago, May 21-23, 1999, and at the Philosophy of Mathematics Conference at the University of California, Santa Barbara, Feb. 4-6, 2000, and especially to Stewart Shapiro and Tony Anderson, for helpful comments on earlier drafts of this paper. (shrink)

Volume 19, Issue 7, July 2019, Page 66-67.