In lieu of an abstract, here is a brief excerpt of the content:Buddhist-Christian Studies 20 (2000) 282-287 [Access article in PDF] Book Review Wrestling with the Ox: A Theology of Religious Experience Wrestling with the Ox: A Theology of Religious Experience. By Paul O. Ingram. New York: Continuum, 1997. 276 pp. Paul Ingram has set out a formidable task for himself. Even though he identifies himself as an historian of religion, he has chosen to push beyond phenomenological description of the (...) religious enterprise to make a prescriptive proposal, a theology that can illuminate and guide the great faith traditions. Simultaneously, his proposal aims at interpreting the necessity and the outlines of interreligious dialogue. Such dialogue, he argues, is not a peripheral enterprise reserved for a handful of religious exotics. It is central to understanding any and all religious traditions in the postmodern world.All religions have been discovered to reflect inescapable historical and cultural relativization. The old assumptions of unconditional claims and absolute assertions are no longer tenable. The world shared by all the religions today is characterized by an unavoidable religious pluralism. It is in this context that religious affirmations must be expressed and religious life embraced. According to Ingram, this is no accident. It is an ontological inevitability. As process philosophy and its roots in modern physics [End Page 282] --and, interestingly, the Buddhist recognition of dependent co-arising (pratitya samutpada)--understand, reality is unending process. This entails the inevitable relativization and contextualization of everything--including the religions. The theological questions that emerge are crucial ones. Among them: Is there an ultimate unity underlying this diversity, generic features common to all conditioned forms of religious faith and practice? What does religious pluralism show about the nature of the Sacred and of being human? Is any one religious Way best or is this even a question worth pursuing?Ingram proposes that the answers to these questions and the only viable response to religious pluralism can be found in interreligious dialogue. He does not even consider religious exclusivism as a viable option. This means only "inclusivist" and "pluralist" approaches are worthy of consideration, approaches that he names the "theology-of-religions model" and the "primordial model" respectively. Both recognize the problem of the relativity of belief and seek to overcome it. Both see the importance of interreligious dialogue as fundamental to religious affirmation. But Ingram rejects the theology-of-religions model because, despite its openness to other religions and its willingness to have interreligious encounter shape the affirmations and life of the believing community, it is self-encapsulated. One moves out from a particular community of faith, "passing over"--to use the language of John Dunne--into another religious Way in order to be creatively transformed, but only to return once again to the original community of faith. One stays within "the theological circle" (Tillich), one does not get beyond it. Some Christians, accordingly, make a blanket assertion about the ultimate and unsurpassable lordship of Christ despite their indebtedness to the creative transformation that interreligious dialogue can produce.Ingram clearly prefers the "primordial model" even though he recognizes that some of its classical proponents have made it vulnerable to criticism. Aldous Huxley's "perennial philosophy" and Sri Ramakrishan's assertion that "all paths lead to the summit" will not do. Huxley assumes that the mystical recognition of the ineffable common root of all religion critiques the historical and culturally relative expressions of this ultimate reality. But Huxley's attempt to make an end run around the problems involved won't work. His approach is ahistorical, makes uncritical assumptions about the nature of mysticism and religious experience, and is totally undialogical. The particularities of the differing religious Ways are dissolved in a universal solvent that is anything but self-evident. The same holds true for Ramakrishnan's subtly imperialistic (and very Hindu) model. It too has no room for religious particularity and imposes a pre-conceived common denominator on the multiplicity of religious phenomena.Ingram derives his term "primordial model" from Huston Smith's argument that a "primordial tradition" can be discerned underlying the multiplicity and variety of religious expressions. His is... (shrink)

In the old days, material egalitarians tended to favor equality of outcome advantage, on some suitable conception of advantage. Under the influence of Dworkin’s seminal articles on equality, contemporary material egalitarians have tended to favor equality of brute luck advantage---on the grounds that this permits people to be held appropriately accountable for the benefits and burdens of their choices. I shall argue, however, that a plausible conception of egalitarian justice requires neither that brute luck advantage always be equalized (...) nor that people always bear the full cost of their voluntary choices. Instead, justice requires that initial opportunities for advantage be equalized---roughly along the lines suggested by Arneson and Cohen. Brute luck egalitarianism and initial opportunity egalitarianism are fairly similar in motivation, and as a result they have not been adequately distinguished. Once the two views are more clearly contrasted, equality of opportunity for advantage will, I claim, be seen to be a more plausible conception of equality. (shrink)

The structure of our ethical experience depends, crucially, on a fundamental distinction between what we are responsible for doing or deciding and what is given to us. As such, the boundary between chance and choice is the spine of our conventional morality, and any serious shift in that boundary is thoroughly dislocating. Against this background, I analyze the way in which techniques of prenatal genetic diagnosis (PGD) pose such a fundamental challenge to our conventional ideas of justice and moral responsibility. (...) After a short description of the situation, I first examine the influential luck egalitarian theory of justice, which is based on the distinction between choice and luck or, more specifically, between option luck and brute luck, and the way in which it would approach PGD (section II), followed by an analysis of the conceptual incoherencies (in section III) and moral problems (in section IV) that come with such an approach. Put shortly, the case of PGD shows that the luck egalitarian approach fails to express equal respect for the individual choices of people. The paradox of the matter is that by overemphasizing the fact of choice as such, without regard for the social framework in which they are being made, or for the fundamental and existential nature of particular choices—like choosing to have children and not to undergo PGD or not to abort a handicapped fetus—such choices actually become impossible. (shrink)

According to luck egalitarianism, inequalities are justified if and only if they arise from choices for which it is reasonable to hold agents responsible. This position has been criticised for its purported harshness in responding to the plight of individuals who, through their own choices, end up destitute. This paper aims to assess the Harshness Objection. I put forward a version of the objection that has been qualified to take into account some of the more subtle elements of the (...) luck egalitarian approach. Revising the objection in this way suggests that the Harshness Objection has been overstated by its proponents: because luck egalitarians are sensitive to the influence of unequal brute luck on individuals’ choices, it is unlikely that there will be any real world cases in which the luck egalitarian would not have to provide at least partial compensation. However, the Harshness Objection still poses problems for the luck egalitarian. First, it is not clear that partial compensation will be sufficient to avoid catastrophic outcomes. Second, the Harshness Objection raises a theoretical problem in that a consistent luck egalitarian will have to regard it as unjust if any assistance is provided to the victim of pure option luck, even if such assistance could be provided at no cost. I consider three strategies the luck egalitarian could pursue to accommodate these concerns and conclude that none of these strategies can be maintained without either violating basic luck egalitarian principles or infringing upon individual liberty. (shrink)

:The distinction between brute luck and option luck is fundamental for luck egalitarianism. Many luck egalitarians write as if it could be used to specify which outcomes people should be held responsible for. In this paper, I argue that the distinction can’t be used this way. In fact, luck egalitarians tend to rely instead on rough intuitive judgements about individual responsibility. This makes their view vulnerable to what’s known as the neutrality objection. I show (...) that attempts to avoid this objection are unsuccessful. I conclude that until it provides a better account of attributing responsibility, luck egalitarianism remains incomplete. (shrink)

Egalitarian thinkers have adopted Ronald Dworkin’s distinction between brute and option luck in their attempts to construct theories that better respect our intuitions about what it is that egalitarian justice should equalize. I argue that when there is no risk-free choice available, it is less straightforward than commonly assumed to draw this distinction in a way that makes brute-luck egalitarianism plausible. I propose an extension of the brute-luck–option-luck distinction to this more general case. The (...) generalized distinction, called the ‘least risky prospect view’ of brute luck, implies more redistribution than Dworkin’s own solution (although less than called for by some of his other critics). Moreover, the generalized brute-luck–option-luck distinction must be parasitical on an underlying non-egalitarian theory of which sets of options are reasonable. The presupposed prior theory may be inimical to the claim that justice requires equality rather than some other distributive pattern. (shrink)

'Moral luck' refers to the phenomenon whereby one's degree of blameworthiness for what one has done varies on account of factors beyond one's control. Applying concepts of Dworkin's from the domain of distributive justice, I draw a distinction between 'option moral luck,' which is that to which one has exposed oneself as the result of one's voluntary choices, and 'brute moral luck,' which is that which is unchosen and unavoidable. I argue that option moral (...) class='Hi'>luck is not ruled out on grounds of unfairness. I also offer a non-fairness-based rejection of brute moral luck and defense of option moral luck. (shrink)

Critics of luck egalitarianism have claimed that, far from providing a justification for the public insurance functions of a welfare state as its proponents claim, the view objectionably abandons those who are deemed responsible for their dire straits. This article considers seven arguments that can be made in response to this ‘abandonment objection’. Four of these arguments are found wanting, with a recurrent problem being their reliance on a dubious sufficientarian or quasi-sufficientarian commitment to provide a threshold of goods (...) unconditionally. Three arguments succeed, showing that luck egalitarians have good reasons for assisting ‘negligent victims’ on account of changes that may occur in an individual between the time of their choice and their subsequent disadvantage, bad option luck, and doubts about free will and responsibility. Luck egalitarianism is therefore shown to offer strong support for public insurance. (shrink)

Luck egalitarianism is a family of egalitarian theories of distributive justice that aim to counteract the distributive effects of luck. This article explains luck egalitarianism's main ideas, and the debates that have accompanied its rise to prominence. There are two main parts to the discussion. The first part sets out three key moves in the influential early statements of Dworkin, Arneson, and Cohen: the brute luck/option luck distinction, the specification of brute luck in (...) everyday or theoretical terms and the specification of advantage as resources, welfare, or some combination of these. The second part covers three later developments: the democratic egalitarian critique of luck egalitarianism, the luck egalitarian acceptance of pluralism, and luck egalitarian doubts about the significance of the brute luck/option luck distinction. (shrink)

abstract This article defends luck egalitarianism as an interpretation of the egalitarian ideal against two major criticisms levelled against it by Elizabeth Anderson — that it is trapped in the distributive paradigm, and that it treats the victims of bad option luck too harshly to be considered an egalitarian theory. Against the first criticism, I argue that luck egalitarianism will condemn non‐material inequalities and injustices if an appropriate conception of well‐being is adopted. I demonstrate this by (...) showing how the approach is sensitive to the five faces of oppression developed by Iris Young. Although the second criticism is more troubling, it does not defeat luck egalitarianism, either. I will show that few of the inequalities that arise in the real world result from option luck. Further, if cases do occur, rather than abandoning the theory, the best response is to combine luck egalitarianism with another egalitarian principle that will ensure that the basic needs of all citizens are satisfied. The paper concludes by defending the appeal of the distinction between option luck and brute luck, in light of the preceding discussion. (shrink)

In this article, we examine how language and linguistic membership might feature in luck egalitarianism, what a luck-egalitarian theory of linguistic justice would look like, and, finally, what the emphasis on language teaches us about the validity of standard luck-egalitarian assumptions. We show that belonging to one language group rather than another is a morally arbitrary feature and that where membership of a specific linguistic group affects individual chances, the effects of such bad brute luck ought (...) to be neutralized on the luck-egalitarian view. We assess two ways of redressing those kinds of unjustified inequalities: the ‘universal language’ option and the ‘linguistic advantages for all’ option. But we also argue, in the second part, that exploring luck egalitarianism through the lens of language exposes some difficulties intrinsic in many existent luck-egalitarian theories. We argue that treating circumstances one identifies with as choices is problematic. In addition, we argue that the linguistic preconditions of both the capacity to be responsible as well the exercise of responsibility complicate the idea of individual responsibility on which most luck-egalitarian theories rely. We conclude by suggesting the need to develop a luck-egalitarian theory of justice which is less reliant on causal features of the distinction between choice and circumstance and which is more sensitive to the idea of collective cooperation as opposed to individual responsibility. (shrink)

This article explores the Rawlsian goal of ensuring that distributions are not influenced by the morally arbitrary. It does so by bringing discussions of distributive justice into contact with the debate over moral luck initiated by Williams and Nagel. Rawls’ own justice as fairness appears to be incompatible with the arbitrariness commitment, as it creates some equalities arbitrarily. A major rival, Dworkin’s version of brute luck egalitarianism, aims to be continuous with ordinary ethics, and so is (a) sensitive (...) to non-philosophical beliefs about free will and responsibility, and (b) allows inequalities to arise on the basis of option luck. But Dworkin does not present convincing reasons in support of continuity, and there are compelling moral reasons for justice to be sensitive to the best philosophical account of free will and responsibility, as is proposed by the revised brute luck egalitarianism of Arneson and Cohen. While Dworkinian brute luck egalitarianism admits three sorts of morally arbitrary disadvantaging which correspond to three forms of moral luck (constitutive, circumstantial, and option luck), revised brute luck egalitarianism does not disadvantage on the basis of constitutive or circumstantial luck. But it is not as sensitive to responsibility as it needs to be to fully extinguish the influence of the morally arbitrary, for persons under it may exercise their responsibility equivalently yet end up with different outcomes on account of option luck. It is concluded that egalitarians should deny the existence of distributive luck, which is luck in the levels of advantage that individuals are due. (shrink)

A situation in which luck determines what happens in our lives is composed of two elements: the existence of a multiplicity of possible outcomes, and lack of control over the situation, namely that we have no way, or at least no meaningful way, to affect the outcome. Adjudication is a luck situation: law is indeterminate and in a decent society litigants are not supposed to have control over their judges. Can we minimize luck in adjudication? The primary (...) way to do this is to make every decision given in the system resemble as much as possible the decision that would have been reached by the majority of the judges active in the system had they all been given the opportunity to decide the case at hand as a group. Put differently, luck in judicial decisions can be lessened by increasing the number of judges in panels. Even though in a decent society the option of controlling the conduct of judges is out of the question, several means do exist for litigants to employ some control over the identity of the persons who are to resolve their disputes. (shrink)

Luck egalitarianism is the view that equality requires the influence of luck on distributive outcomes to be neutralized. The standard version of the view, brute-luck egalitarianism, neutralizes brute luck (the upshot of non-declinable risks) while allowing option luck (the upshot of declinable risks) to stand. This article argues that this view should be rejected in favour of all-luck egalitarianism, which neutralizes brute luck and option luck alike. There are three parts (...) to this overall argument. The first shows that brute-luck egalitarianism’s reasons for neutralizing brute luck’s distributive effects should lead it to also neutralize a subset of option luck. The second demonstrates that various revisions of brute-luck egalitarianism such that it neutralizes this subset (and related ones) are ultimately unsuccessful. The third defends the remaining option of neutralizing option luck generally, as all-luck egalitarianism proposes. (shrink)

Luck egalitarianism assigns a central role to personal responsibility in egalitarian justice. In the context of healthcare, luck egalitarianism is the view that the distribution of medical and healthcare resources—or common resources in general—should respond to the (im)prudence of individuals. Recently, Joar Björk, Gert Helgesson, and Niklas Juth have argued that it is impractical to use luck egalitarianism as a normative framework in healthcare because it has no reasonable way of dealing with the imprudent. In response to (...) their argument, this paper first suggests that the epistemic problems of applying luck egalitarianism to the healthcare context raised by Björk et al. can be circumvented by using the exemption system as a policy application of luck egalitarian healthcare justice. This paper then shows that an _ex ante_ policy, a tax system with refunds, can reasonably be adopted as a luck egalitarian institutional design of healthcare policy. We argue that the proposed conception of luck egalitarianism can deal with the problem of differential option luck. Finally, we show that the threshold problem for the definition of imprudence does not refute the proposed _ex ante_ policy. (shrink)

Many philosophers believe that luck egalitarianism captures “desert-like” intuitions about justice. Some even think that luck egalitariansm distributes goods in accordance with desert. In this paper, we argue that this is wrong. Desertism conflicts with luck egalitarianism in three important contexts, and, in these contexts, desertism renders the proper moral judgment. First, compared to desertism, luck egalitarianism is sometimes too stingy: it fails to justly compensate people for their socially valuable contributions—when those contributions arose from “ (...) class='Hi'>option luck”. Second, luck egalitarianism is sometimes too restrictive: it fails to justly compensate people who make a social contribution when that contribution arose from “brute luck”. Third, luck egalitarianism is too limited in scope: it cannot diagnose economic injustice arising independently of comparative levels of justice. The lesson of this paper is that luck egalitarians should consider supplementing their theory with desert considerations. Or, even better, consider desertism as a superior alternative to their theory. (shrink)

abstract We seem to have conflicting intuitions regarding luck and war, and we seem to be faced with a dilemma. Either, we deny that a war can be made just or unjust as a result of luck, and we accept that we should not appeal to the outcome when claiming that the war was or was not justified. Or, alternatively, we allow that it is legitimate to base our judgements on the outcome, but as a result we must (...) accept that luck can make a war just or unjust. Traditionally, these have been taken to be the two forks of the dilemma, but, in this paper, I argue that they are not the only options. Rather, we can appeal to the outcome of our actions without claiming that this is, in anyway, an appeal to moral luck. Rather, the outcome provides us with evidence. (shrink)

There is a large and growing body of empirical work on people's intuitions about distributive justice. In this paper, we investigate how well luck egalitarianism and desertism--the two normative approaches that appear to cohere well with people's intuitions--are supported by more fine-grained findings in the empirical literature. The time is ripe for a study of this sort, as the positive literature on justice has blossomed over the last three decades. The results of our investigation are surprising. In three different (...) contexts (good option luck, good brute luck, and bad brute luck) in which the demands of luck egalitarianism and those of a mainstream desert-based view come apart, the latter carries the day. One ramification of these findings is that people's intuitions about justice are moralized; that is, they appeal to particular conceptions of the good. Luck egalitarians must decide whether to embrace these moralized intuitions by adopting desertism--or to resist them. (shrink)

The luck egalitarian view famously maintains that inequalities in individuals’ circumstances are unfair or unjust, whereas inequalities traceable to individuals’ own responsible choices are fair or just. On this basis, the distinction between so-called brute luck and option luck has been seen as central to luck egalitarianism. Luck egalitarianism is interpreted, by advocates and opponents alike, as a view that condemns inequalities in brute luck but permits inequalities in option luck. It (...) is also thought to be expressed in terms of the view that no individual ought to be worse off other than because of a fault or choice of his or her own. I argue that these two characterizations of luck egalitarianism are not equivalent and that, properly understood, luck egalitarianism is compatible with widespread, potentially radical, inequalities in brute luck. (shrink)

Luck egalitarians typically hold that it is bad for some to be worse off than others through no fault or choice of their own. In this paper I want to address two complaints against standard luck egalitarianism that do not question responsibility-sensitivity. The first objection says that equality itself lacks inherent non-instrumental value, and so the luckist component ought to be attached to a different pattern, say prioritarianism. The second objection also endorses luckism but worries that luck (...) egalitarianism as conventionally formulated is committed in fact to neutralizing not just brute luck but also option luck. And this would mean, among other things, compensating unsuccessful gamblers, which is something many egalitarians find counterintuitive. My aim is to show that there is a way for luck egalitarianism to meet both criticisms; that it can maintain its egalitarian credentials while avoiding the counterintuitive consequences of compensating unsuccessful gamblers. To do so, I propose, we ought to understand luck egalitarianism as resting on the disvalue of being arbitrarily worse off compared to others. In addition, I suggest, the badness of luck egalitarian inequality – that of arbitrary disadvantage – has both an inter-personal and an intra-personal dimension. (shrink)

This article raises a puzzle about luck and meaning in life. The puzzle shows that, in certain cases involving luck, standard intuitions about the meaningfulness of various lives conflict with basic theoretical assumptions about the nature of meaning. After setting out the puzzle, several options for resolving it are developed and evaluated.

Modal epistemologists parse modal conditions on knowledge in terms of metaphysical possibilities or ways the world might have been. This is problematic. Understanding modal conditions on knowledge this way has made modal epistemology, as currently worked out, unable to account for epistemic luck in the case of necessary truths, and unable to characterise widely discussed issues such as the problem of religious diversity and the perceived epistemological problem with knowledge of abstract objects. Moreover, there is reason to think that (...) this is a congenital defect of orthodox modal epistemology. This way of characterising modal epistemology is however optional. It is shown that one can non-circularly characterise modal conditions on knowledge in terms of epistemic possibilities, or ways the world might be for the target agent. Characterising the anti-luck condition in terms of epistemic possibilities removes the impediment to understanding epistemic luck in the case of necessary truths and opens the door to using these conditions to shed new light on some longstanding epistemological problems. (shrink)

Most forms of egalitarian theory impose on government to redress the inequalities of fortune that result from bad luck. This Article takes issue with the various forms of this large claim, and argues that decentralized forms of assistance are likely in the long run to do better by the very standards by which egalitarians justify their own program. The alleviation of poverty depends in the first instance on increases in wealth that can only come through private innovation and technological (...) advances. These have in fact produced major improvements in overall well-being, with disproportionate advances for the poor. But if one starts with Dworkin’s unsustainable distinction between option and brute luck, or Nussbaum and Sen’s capability theory, then no egalitarian theory can deliver on the promise to level differences in wealth without seriously compromising overall levels of social welfare. By expanding the scope of government regulation, these proposals open the door to selfish political forces whose political clout ensures that ill-conceived programs, such as the Americans with Disabilities Act, frustrate the very goals they hope to achieve. State intervention to redistribute resources should be understood as a last resort for dealing with problems of ill fortune. (shrink)

Explores the embodied aspects of ownership and private property as these emerge in a range of American literary texts across the late eighteenth and early nineteenth century.

Explores the embodied aspects of ownership and private property as these emerge in a range of American literary texts across the late eighteenth and early nineteenth century.

We prove that forp-optimal fields a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is semialgebraic. Combining (...) the two we get a preparation theorem for definable functions onp-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension. (shrink)

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, (...) from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators. (shrink)

In this paper a new resolution to the gamer’s dilemma is presented. The first part of the paper is devoted to strictly formulating the dilemma, and the second to establishing its resolution. The proposed resolution, the grave resolution, aims to resolve not only the gamer’s dilemma, but also a wider set of analogous paradoxes – which together make up the paradox of treating wrongdoing lightly.

Most people agree that murder is wrong. Yet, within computer games virtual murder scarcely raises an eyebrow. In one respect this is hardly surprising, as no one is actually murdered within a computer game. A virtual murder, some might argue, is no more unethical than taking a pawn in a game of chess. However, if no actual children are abused in acts of virtual paedophilia (life-like simulations of the actual practice), does that mean we should disregard these acts with the (...) same abandon we do virtual murder? In this paper I shall outline several arguments which attempt to permit virtual murder, whilst prohibiting virtual paedophilia. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [math] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, (...) and contains all large cardinals that can be characterized through the validity of the principle [math] for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak [math]-indescribability, can be canonically characterized through localized versions of the principle [math]. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak [math]-indescribability. (shrink)

Let k be an infinite cardinal. A subset of $(^k k)^n $ is a $\Sigma _1^1 $ -subset if it is the projection p[T] of all cofinal branches through a subtree T of $(lt;kk)^{n + 1} $ of height k. We define $\Sigma _k^1 - ,\Pi _k^1 $ - and $\Delta _k^1$ subsets of $(^k k)^n $ as usual. Given an uncountable regular cardinal k with k = k (...) there is a shrink)

In this paper I will evaluate Ali’s dissolution of the gamer’s dilemma. To this end the dilemma will be summarized and Ali’s dissolution formulated. I conclude that Ali has not dissolved the dilemma (at least not fully).

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals κ with the property that the collection of all initial segments of the wellordering is definable by a Σ1‐formula with parameter κ. A short argument shows that the existence of a measurable cardinal δ implies that such wellorderings do not exist at δ‐inaccessible cardinals of cofinality not equal to δ and their successors. In contrast, our main result shows that (...) these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that imposes restrictions on the existence of such wellorderings at uncountable cardinals. Finally, we generalise the above result to the minimal model containing two measurable cardinals. (shrink)

Russell and Walker proposed different ways of constructing instants from events. For an explanation of "time as a continuum," Thomason favored Walker's construction. The present article shows that Russell's construction fares as well. To this end, a mathematical characterization problem is solved which corresponds to the characterization problem that Thomason solved with regard to Walker's construction. It is shown how to characterize those event structures (formally, interval orders) which, through Russell's construction of instants, become linear orders isomorphic to a given (...) (or, deriving, to some—nontrivial ordered) real interval. As tools, separate characterizations for each of resulting (i) Dedekind completeness, (ii) separability, (iii) plurality of elements, (iv) existence of certain endpoints are provided. Denseness is characterized to replace Russell's erroneous attempt. Somewhat minimal nonconstructive principles needed are exhibited, and some alternative approaches are surveyed. (shrink)

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or (...) an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text]. (shrink)

Let p be prime number, K be a p‐adically closed field, a semi‐algebraic set defined over K and the lattice of semi‐algebraic subsets of X which are closed in X. We prove that the complete theory of eliminates quantifiers in a certain language, the ‐structure on being an extension by definition of the lattice structure. Moreover it is decidable, contrary to what happens over a real closed field for. We classify these ‐structures up to elementary equivalence, and get in particular (...) that the complete theory of only depends on m, not on K nor even on p. As an application we obtain a classification of semi‐algebraic sets over countable p‐adically closed fields up to so‐called “pre‐algebraic” homeomorphisms. (shrink)