In the massive plot twist at the end of BioShock Infinite, the writers beautifully put forth a hypothesis that individuals might exist in more than one possible world. In philosophy, the idea that an individual can exist in more than one world is called transworld identity. An important rival to transworld identity theory is counterpart theory, the idea that individuals cannot exist in more than one possible world and are therefore “world bound.” Modal realism is the thesis according (...) to which there is an infinite number of possible worlds that really exist in the most robust sense, relative to themselves. The world of Columbia is one possible world. The world of Rapture is another possible world. Both the actualist and the possibilist agree on these claims. Most important to the themes present in BioShock, transworld individuality has profound implications for differing notions of freedom. (shrink)

BioShock Infinite is a piece of fiction that lets one peer into a world where this linearity seems overridden by a multiverse where all the possibilities exist. Stories are important for video games. Its story is one of the reasons BioShock Infinite resonates with audiences all around the world. The field of philosophy that deals with art is called aesthetics. If one think that it's even worth asking the question of whether BioShock Infinite is (...) art or not, then one might want to consider this: What is it we're asking about in this case? When we ask whether Infinite is art, we're asking about the experience of playing it. BioShock Infinite is still one of the greatest video game stories and worlds of all time. In this, it represents a big step towards artful video game design. BioShock succeeds in showing that the possibilities for artful storytelling are infinite. (shrink)

BioShock Infinite begins with the question of “founding.” One enters Columbia for the first time on “Secession Day,” the anniversary of Columbia's secession from the United States in 1902, and the commemoration of the founding of Columbia as the “New Eden.” Racial difference is one of the major antagonisms in BioShock Infinite. BioShock Infinite exemplifies Carl Schmitt's concept of the political, as grounded on fundamental antagonisms that express the will of “the people” of Columbia. (...) Sovereignty is the power to decide what one will do outside of any rules or laws. The questions of decision and responsibility play a key role in the story and gameplay of BioShock Infinite. Players are forced to make a number of decisions throughout BioShock Infinite, without any way of knowing what the outcomes of these decisions will be. It turns out that none of these decisions has any real impact on the game. (shrink)

At the end of BioShock Infinite, Booker is faced with the challenge of not allowing the tragedy to befall Columbia. There has to be a way, he believes, to prevent the rise of Father Comstock, the imprisonment and abuse of Elizabeth, and the creation of a Columbia that persecutes people for both religious and racial reasons. Booker's action is predicated on the necessity of Booker becoming Comstock. Elizabeth takes Booker back to Father Comstock's creation and it is revealed (...) that Comstock's coming to power took place when DeWitt accepted Christian baptism after his participation in the Wounded Knee massacre. The notion that Booker DeWitt had to become Father Comstock because of his baptism portrays a God that is overly deterministic, however. In reality, God may be omnipotent and omniscient, and yet humans may be free. (shrink)

The story of the Vox Populi embodies conflict theory, one popular interpretation of Marx's ideas, portraying a bloody revolution that loses sight of its ideals, turns anarchistic, and becomes the new oppressor. In Columbia, Zachary Hale Comstock and Jeremiah Fink illustrate the way the bourgeoisie may come to create and control the means of production. As the friction builds between the bourgeoisie and the proletariat, historical processes contribute to the inevitable collapse of capitalism. In BioShock Infinite, the simmering (...) friction between the upper and lower classes boils over into the rise of the Vox Populi. The functionalist perspective often casts religion as a social structure that keeps society together despite any negative costs it may have for any particular group. In order for the bourgeoisie to own the means of production and keep the working class oppressed, it uses propaganda and indoctrination as powerful agents of control. (shrink)

Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only (...) a finite amount of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)

“Father” Zachary Hale Comstock is a self‐professed prophet, religious zealot, and racist, who has kept his “heir” under lock and key in the floating city of Columbia. Booker DeWitt is a washed‐up, disgraced ex‐Pinkerton agent haunted by his participation in the Wounded Knee Massacre. He enters Columbia to rescue Elizabeth in exchange for having his gambling debts settled. After much bloodshed, Booker saves Elizabeth and kills Comstock. In the past, Booker attended a baptism to assuage his guilt over Wounded Knee. (...) In some worlds he rejected the baptism, got married, had a daughter, Anna, and ended up as an indebted alcoholic ex‐Pinkerton agent. In the Comstock world, Anna is renamed Elizabeth in order to conceal her true origins. BioShock Infinite paints a metaphysical picture that treats the many‐worlds interpretation of quantum mechanics as reality. (shrink)

BioShock Infinite follows the journey of Booker DeWitt, a reluctant detective from New York City who is transplanted via rowboat and lighthouse‐turned‐rocket to the city of Columbia. Booker is charged with the task of bringing back a girl in order to wipe away his debt. The girl, Elizabeth Comstock, has been locked in a tower since infancy by her father, Zachary Comstock, and is protected by a menacing Songbird. Although the game centers on Booker and his story, this (...) chapter focuses on Elizabeth, looking at her through a feminist lens. Freeing Elizabeth from her tower, one could argue, is in a way freeing her from oppression of men. If we take Elizabeth as her own person, rather than as a mere companion, does Elizabeth's freedom from tower mean real freedom for her? Does Booker treat Elizabeth any differently than Comstock does? And, what really is Elizabeth's role in Booker's drowning?. (shrink)

If one knows everything there is to know about a person's state while playing BioShock, if one could describe the exact neurons firing, the synapses responding, and so on, one still wouldn't know what it is like to play BioShock. This is precisely what phenomenology is interested in. Some philosophers would argue that there are a great number of phenomenological methods, but the author thinks they basically fall into two categories: the phenomenological tradition and art as phenomenology. (...) class='Hi'>Infinite is phenomenologically interesting because the game's designers have poured phenomenological ideas into Columbia and Elizabeth, which one can then read as if they were books containing a phenomenological analysis of reality. Once the player is comfortable in the role of protagonist, one is easily absorbed by the worlds of Rapture and Columbia. The phenomenological relevance of these insights is that they reveal something about oneself. (shrink)

BioShock begins simply with the text “1960 Mid‐Atlantic.” The player's horizon shifts to accommodate this fact, like not being so surprised that Jack can smoke in the airplane. What follows in BioShock is the development of a narrative where it is assumed that Jack is entering Rapture for the first time in his life. Later, it is revealed that he is not. When Andrew Ryan exposes Jack's real identity, Ryan is falsifying both the narrative of Jack coming to (...) Rapture for the first time and the meta‐narrative of the player operating a free agent as an avatar. Understanding how BioShock effectively manipulates players will take us through a variety of territories: cognitive science, philosophy of mind, philosophical hermeneutics, philosophy of video gaming, and philosophy of free will. It's all a testament to the brilliance of BioShock and a demonstration of how video games can teach us‐ even change us. (shrink)

Everyone misses the point of Plato's cave. What a coincidence, because everyone also misses the point of BioShock. The moment one's interactivity with the game is revealed as a fake isn't the moment when one kills Andrew Ryan in a cutscene. It's what happens after that. Atlas tells to abort the self‐destruct sequence. One has the choice of whether to abort self‐destruct sequence or not, but, positioned as it is, that choice has been exposed as meaningless within the basic (...) fabric of game's mechanics. The non‐choice of disarming self‐destruct sequence ends up in the same place as the other, more famous ethical dilemma of BioShock; the choice of whether to harvest the Little Sisters or to save them. That's the part that everyone gets wrong where BioShock is concerned, and so it's the part where comparing the game to Plato's cave can really help get both of them right. (shrink)

I will explore the paradoxical nature of epistemic access. By critiquing the traditional conception of mental states that are labelled as ’knowledge’, I demonstrate the susceptibility of these states to an infinite regress, thus, challenging their existence and validity. I scrutinise the assumption that an epistemic agent can have complete epistemic access to all facts about a given object while simultaneously being ignorant of certain truths that impact the very knowledge claims about the object. I further analyse the implications (...) of this paradox when attempting to fine-grain the parameters of epistemic access, which happens to reveal the vacuity of such definitions and the inevitability of the regress. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Bioshock and the Art of RaptureGrant TavinorI am Andrew Ryan, and I am here to ask you a question. Is a man not entitled to the sweat of his brow? "No!" says the man in Washington, "It belongs to the poor." "No!" says the man in the Vatican, "It belongs to God." "No!" says the man in Moscow, "It belongs to everyone." I rejected these answers; instead, I (...) chose something different. I chose the impossible. I chose Rapture! A city where the artist would not fear the censor; where the scientist would not be bound by petty morality; where the great would not be constrained by the small! And with the sweat of your brow, Rapture can become your city as well.Andrew Ryan, opening BioshockIBioshock is the Masterpiece of recent gaming. Genre-wise the game is a first person-shooter survival-horror game—already a complicated mix of gaming forms—and Bioshock is excellent in these gaming terms: playing the game is a consistently engaging, challenging, and tense experience. The narrative of the game, set in Rapture, a dystopian city beneath the sea, fits perfectly with this interactive gaming form. The premise of Bioshock is a parody of Ayn Rand's objectivist novel Atlas Shrugged. Led by the magnate Andrew Ryan, the industrialists, artists, and scientists have retreated from the world and built Rapture through sheer force of will. But when the player enters the city at the beginning of the game, it is evident that nature is pushing back, and that the sea is slowly but surely retaking Rapture, while its citizens have become corrupted by the arts and sciences the city was built upon: as genetically manipulated "splicers" they now creep through Rapture's darkened halls bemoaning their lost humanity. Andrew Ryan "chose the impossible," [End Page 91] and inevitably his hubris is being repaid by the recalcitrance of human nature and the impermanence of human achievement. Much of the immense fun of Bioshock derives from the irony of an objectivist utopia running amok.The immediate visual impact of Bioshock, depending on its employment of cutting-edge high definition 3D computer graphics, is striking. The aesthetic qualities of the water, in particular, have drawn wide praise. The world of Rapture is also presented through an engaging and excellent style. To depict this decaying world, Bioshock draws on the architectural motifs and cultural themes of 1930s and 1940s America. Portrayals of decaying art deco facades, faded Hollywood socialites, and echoes of Hearst, Hughes, and Citizen Kane, are combined with period music and philosophical and literary references to produce a coherent artistic statement.Two features of Bioshock bear out its unique artfulness as a game. First is the role in the game-world of the Little Sisters: cute little girls that pose a game obstacle with which the player must decide how to deal. In Bioshock, the task of harvesting the game-world resources needed to survive is given a moral dimension in that the resources can be extracted only from the Little Sisters by killing them, and so a key part of the gameplay rests on an emotionally provocative moral choice. The second artful aspect of the game that warrants special attention is a narrative twist partway through the game where it becomes evident that player's principle sympathetic source of information in the game-world is a dissemblance. Bioshock has an interactive twist on an untrustworthy narrator, which clouds and complicates the decisions the player has made in Rapture. Rather than an actor, the player-character is a pawn in someone else's game. At the forefront of both of these aspects of Bioshock are considerations of freewill and morality.To really appreciate the art of Bioshock we need to understand its nature as a game, and moreover, to treat the gaming nature seriously and with sympathy. Bioshock is a videogame, and it is also clearly art; but it is not as if the game is art despite its being a videogame and that the art is a mere gloss or veneer. Rather, Bioshock's nature as a game allows it to be art of a distinctive kind. Treating videogames as a kind... (shrink)

Considered a sign of the ‘coming of age’ of video games as an artistic medium, the award-winning BioShock franchise covers vast philosophical ground. _BioShock and Philosophy: Irrational Game, Rational Book _presents expert reflections by philosophers on this critically acclaimed and immersive fan-favorite. Reveals the philosophical questions raised through the artistic complexity, compelling characters and absorbing plots of this ground-breaking first-person shooter Explores what _BioShock_ teaches the gamer about gaming, and the aesthetics of video game storytelling Addresses a wide array (...) of topics including Marxism, propaganda, human enhancement technologies, political decision-making, free will, morality, feminism, transworld individuality, and vending machines in the dystopian society of Rapture Considers visionary game developer Ken Levine’s depiction of Ayn Rand’s philosophy, as well as the theories of Aristotle, de Beauvoir, Dewey, Leibniz, Marx, Plato, and others from the Hall of Philosophical Heroes. (shrink)

Suppose you found that the universe around you was infinite—that it extended infinitely far in space or in time and, as a result, contained infinitely many persons. How should this change your moral decision-making? Radically, it seems, according to some philosophers. According to various recent arguments, any moral theory that is ’minimally aggregative’ will deliver absurd judgements in practice if the universe is (even remotely likely to be) infinite. This seems like sound justification for abandoning any such theory. (...) -/- My goal in this thesis is simple: to demonstrate that we need not abandon minimally aggregative theories, even if we happen to live in an infinite universe. I develop and motivate an extension of such theories, which delivers plausible judgements in a range of realistic cases. I show that this extended theory can overcome key objections—both old and new—and that it succeeds where other proposals do not. With this proposal in hand, we can indeed retain minimally aggregative theories and continue to make moral decisions based on what will promote the good. (shrink)

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

People with the kind of preferences that give rise to the St. Petersburg paradox are problematic---but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the *Countable Sure Thing Principle*, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we call (...) *Countable Independence*. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have. (shrink)

Infinite Thought brings together a representative selection of the range of Alain Badiou's work, illustrating the power and diversity of his thought.

In formal ontology, infinite regresses are generally considered a bad sign. One debate where such regresses come into play is the debate about fundamentality. Arguments in favour of some type of fundamentalism are many, but they generally share the idea that infinite chains of ontological dependence must be ruled out. Some motivations for this view are assessed in this article, with the conclusion that such infinite chains may not always be vicious. Indeed, there may even be room (...) for a type of fundamentalism combined with infinite descent as long as this descent is “boring,” that is, the same structure repeats ad infinitum. A start is made in the article towards a systematic account of this type of infinite descent. The philosophical prospects and scientific tenability of the account are briefly evaluated using an example from physics. (shrink)

For aggregative theories of moral value, it is a challenge to rank worlds that each contain infinitely many valuable events. And, although there are several existing proposals for doing so, few provide a cardinal measure of each world's value. This raises the even greater challenge of ranking lotteries over such worlds—without a cardinal value for each world, we cannot apply expected value theory. How then can we compare such lotteries? To date, we have just one method for doing so (proposed (...) separately by Arntzenius, Bostrom, and Meacham), which is to compare the prospects for value at each individual location, and to then represent and compare lotteries by their expected values at each of those locations. But, as I show here, this approach violates several key principles of decision theory and generates some implausible verdicts. I propose an alternative—one which delivers plausible rankings of lotteries, which is implied by a plausible collection of axioms, and which can be applied alongside almost any ranking of infinite worlds. (shrink)

This book examines the philosophy of the nineteenth-century Indian mystic Sri Ramakrishna and brings him into dialogue with Western philosophers of religion, primarily in the recent analytic tradition. Sri Ramakrishna’s expansive conception of God as the impersonal-personal Infinite Reality, Maharaj argues, opens up an entirely new paradigm for addressing central topics in the philosophy of religion, including divine infinitude, religious diversity, the nature and epistemology of mystical experience, and the problem of evil.

In this paper I argue that the infinite regress of resemblance is vicious in the guise it is given by Russell but that it is virtuous if generated in a (contemporary) trope theoretical framework. To explain why this is so I investigate the infinite regress argument. I find that there is but one interesting and substantial way in which the distinction between vicious and virtuous regresses can be understood: The Dependence Understanding. I argue, furthermore, that to be able (...) to decide whether an infinite regress exhibits a dependence pattern of a vicious or a virtuous kind, facts about the theoretical context in which it is generated become essential. It is precisely because of differences in context that he Russellian resemblance regress is vicious whereas its trope theoretical counterpart is not. (shrink)

Infinite regress arguments play an important role in many distinct philosophical debates. Yet, exactly how they are to be used to demonstrate anything is a matter of serious controversy. In this paper I take up this metaphilosophical debate, and demonstrate how infinite regress arguments can be used for two different purposes: either they can refute a universally quantified proposition (as the Paradox Theory says), or they can demonstrate that a solution never solves a given problem (as the Failure (...) Theory says). In the meantime, I show that Black’s view on infinite regress arguments (1996, this journal) is incomplete, and how his criticism of Passmore can be countered. (shrink)

Infinite in All Directions is a popularized science at its best. In Dyson's view, science and religion are two windows through which we can look out at the world around us. The book is a revised version of a series of the Gifford Lectures under the title "In Praise of Diversity" given at Aberdeen, Scotland. They allowed Dyson the license to express everything in the universe, which he divided into two parts in polished prose: focusing on the diversity of (...) the natural world as the first, and the diversity of human reactions as the second half. Chapter 1 is a brief explanation of Dyson's attitudes toward religion and science. Chapter 2 is a one-hour tour of the universe that emphasizes the diversity of viewpoints from which the universe can be encountered as well as the diversity of objects which it contains. Chapter 3 is concerned with the history of science and describes two contrasting styles in science: one welcoming diversity and the other deploring it. He uses the cities of Manchester and Athens as symbols of these two ways of approaching science. Chapter 4, concerned with the origin of life, describes the ideas of six illustrious scientists who have struggled to understand the nature of life from various points of view. Chapter 5 continues the discussion of the nature and evolution of life. The question of why life characteristically tends toward extremes of diversity remains central in all attempts to understand life's place in the universe. Chapter 6 is an exercise in eschatology, trying to define possible futures for life and for the universe, from here to infinity. In this chapter, Dyson crosses the border between science and science fiction and he frames his speculations in a slightly theological context. (shrink)

This book on infinite regress arguments provides (i) an up-to-date overview of the literature on the topic, (ii) ready-to-use insights for all domains of philosophy, and (iii) two case studies to illustrate these insights in some detail. Infinite regress arguments play an important role in all domains of philosophy. There are infinite regresses of reasons, obligations, rules, and disputes, and all are supposed to have their own moral. Yet most of them are involved in controversy. Hence the (...) question is: what exactly is an infinite regress argument, and when is such an argument a good one? (shrink)

One finds intertwined with ideas at the core of evolutionary theory claims about frequencies in counterfactual and infinitely large populations of organisms, as well as in sets of populations of organisms. One also finds claims about frequencies in counterfactual and infinitely large populations—of events—at the core of an answer to a question concerning the foundations of evolutionary theory. The question is this: To what do the numerical probabilities found throughout evolutionary theory correspond? The answer in question says that evolutionary probabilities (...) are “hypothetical frequencies” (including what are sometimes called “long-run frequencies” and “long-run propensities”). In this paper, I review two arguments against hypothetical frequencies. The arguments have implications for the interpretation of evolutionary probabilities, but more importantly, they seem to raise problems for biologists’ claims about frequencies in counterfactual or infinite populations of organisms and sets of populations of organisms. I argue that when properly understood, claims about frequencies in large and infinite populations of organisms and sets of populations are not threatened by the arguments. Seeing why gives us a clearer understanding of the nature of counterfactual and infinite population claims and probability in evolutionary theory. (shrink)

This paper discusses an infinite regress that looms behind a certain kind of historical explanation. The movement of one barbarian group is often explained by the movement of others, but those movements in turn call for an explanation. While their explanation can again be the movement of yet another group of barbarians, if this sort of explanation does not stop somewhere we are left with an infinite regress of barbarians. While that regress would be vicious, it cannot be (...) accommodated by several general views about what viciousness in infinite regresses amounts to. This example is additional evidence that we should prefer a pluralist approach to infinite regresses. (shrink)

How might we extend aggregative moral theories to compare infinite worlds? In particular, how might we extend them to compare worlds with infinite spatial volume, infinite temporal duration, and infinitely many morally valuable phenomena? When doing so, we face various impossibility results from the existing literature. For instance, the view we adopt can endorse the claim that worlds are made better if we increase the value in every region of space and time, or that they are made (...) better if we increase the value obtained by every person. But they cannot endorse both claims, so we must choose. In this paper I show that, if we choose the latter, our view will face serious problems such as generating incomparability in many realistic cases. Opting instead to endorse the first claim, I articulate and defend a spatiotemporal, expansionist view of infinite aggregation. Spatiotemporal views such as this do face some difficulties, but I show that these can be overcome. With modification, they can provide plausible comparisons in cases that we care about most. (shrink)

Consider an infinite series whose items are each explained by their immediate successor. Does such an infinite explanation explain the whole series or does it leave something to be explained? Hume arguably claimed that it does fully explain the whole series. Leibniz, however, designed a very telling objection against this claim, an objection involving an infinite series of book copies. In this paper, I argue that the Humean claim can, in certain cases, be saved from the Leibnizian (...) “infinite book copies” objection, and that this provides an interesting way to defuse some cosmological arguments for the existence of God and to give a non-theistic but complete explanation of the Universe. In the course of my argumentation, I also show that circular explanations can be “self-explanatory” as well: explaining two items by each other can explain the couple of items tout court. (shrink)

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the (...) Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text]. (shrink)

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

BioShock made a big splash not only for the depth of its subject matter, but also for the way it utilized its video game medium to present its big ideas in a uniquely engaging way. The game weaves many themes into its complicated narrative, complete with shifting identities, science fiction superpowers, and survival‐horror overtones. It is clear that BioShock wants to be taken as a spiritual sequel to Rand's philosophical novel Atlas Shrugged, revealing a possible fate for John (...) Galt's mysterious hidden utopia, sought after for much of the novel but never fully revealed, as the book ends just before Galt discloses his society and its plans for the world. Rand develops her philosophy through fiction because she defines art as a selective re‐creation of reality according to an artist's perspective. That is, a work of art is an artist's take on what it is to be human. (shrink)

This article offers a reconstruction of an argument against infinite regress formulated by Aristotle in Posterior Analytics I 22. I argue against the traditional interpretation of the chapter, according to which singular terms and summa genera, in virtue of having restrict logical roles, provide limits for predicative chains, preventing them from proceeding ad infinitum. As I intend to show, this traditional reading is at odds with some important aspects of Aristotle’s theory of demonstration. More importantly, it fails to explain (...) how his proof is connected to a defence of the existence of ultimate explanations, a connection that must be the case if I 19-22 is advancing a foundationalist way-out to a sceptical challenge raised in I 3. (shrink)

000000001. Introduction Call a theory of the good—be it moral or prudential—aggregative just in case (1) it recognizes local (or location-relative) goodness, and (2) the goodness of states of affairs is based on some aggregation of local goodness. The locations for local goodness might be points or regions in time, space, or space-time; or they might be people, or states of nature.1 Any method of aggregation is allowed: totaling, averaging, measuring the equality of the distribution, measuring the minimum, etc.. Call (...) a theory of the good finitely additive just in case it is aggregative, and for any finite set of locations it aggregates by adding together the goodness at those locations. Standard versions of total utilitarianism typically invoke finitely additive value theories (with people as locations). A puzzle can arise when finitely additive value theories are applied to cases involving an infinite number of locations (people, times, etc.). Suppose, for example, that temporal locations are the locus of value, and that time is discrete, and has no beginning or end.2 How would a finitely additive theory (e.g., a temporal version of total utilitarianism) judge the following two worlds? Goodness at Locations (e.g. times) w1:..., 2, 2, 2, 2, 2, 2, 2, 2, 2, ..... w2:..., 1, 1, 1, 1, 1, 1, 1, 1, 1, ..... Example 1 At each time w1 contains 2 units of goodness and w2 contains only 1. Intuitively, we claim, if the locations are the same in each world, finitely additive theorists will want to claim that w1 is better than w2. But it's not clear how they could coherently hold this view. For using standard mathematics the sum of each is the same infinity, and so there seems to be no basis for claiming that one is better than the other.3 (Appealing to Cantorian infinities is of no help here, since for any Cantorian infinite N, 2xN=1xN.). (shrink)

Once one accepts that certain things metaphysically depend upon, or are metaphysically explained by, other things, it is natural to begin to wonder whether these chains of dependence or explanation must come to an end. This essay surveys the work that has been done on this issue—the issue of grounding and infinite descent. I frame the discussion around two questions: (1) What is infinite descent of ground? and (2) Is infinite descent of ground possible? In addressing the (...) second question, I will consider a number of arguments that have been made for and against the possibility of infinite descent of ground. When relevant, I connect the discussion to two important views about the way reality can be structured by grounding: metaphysical foundationalism and metaphysical infinitism. (shrink)

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

Using only uncontentious principles from the logic of ground I construct an infinitely descending chain of ground without a lower bound. I then compare the construction to the constructions due to Dixon and Rabin and Rabern.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

We can have credences in an infinite number of propositions—that is, our opinion set can be infinite. Accuracy-first epistemologists have devoted themselves to evaluating credal states with the help of the concept of ‘accuracy’. Unfortunately, under several innocuous assumptions, infinite opinion sets yield several undesirable results, some of which are even fatal, to accuracy-first epistemology. Moreover, accuracy-first epistemologists cannot circumvent these difficulties in any standard way. In this regard, we will suggest a non-standard approach, called a relativistic (...) approach, to accuracy-first epistemology and show that such an approach can successfully circumvent undesirable results while having some advantages over the standard approach. (shrink)

This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.

This chapter challenges Cantor’s notion of the ‘power’, or ‘cardinality’, of an infinite set. According to Cantor, two infinite sets have the same cardinality if and only if there is a one-to-one correspondence between them. Cantor showed that there are infinite sets that do not have the same cardinality in this sense. Further, he took this result to show that there are infinite sets of different sizes. This has become the standard understanding of the result. The (...) chapter challenges this, arguing that we have no reason to think there are infinite sets of different sizes. It begins with an initial argument against Cantor’s claim that there are infinite sets of different sizes and then proceeds, by way of an analogy between Cantor’s mathematical result and Russell’s paradox, to a more direct argument. (shrink)