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)

'Chains of Being' argues that there can be infinite chains of dependence or grounding. Cameron also defends the view that there can be circular relations of ontological dependence or grounding, and uses these claims to explore issues in logic and ontology.

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less (...) than the ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

I analyze the role of infinite idealizations used in the renormalization group (RG hereafter) method in explaining universality across microscopically different physical systems in critical phenomena. I argue that despite the reference to infinite limit systems such as systems with infinite correlation lengths during the RG process, the key to explaining universality in critical phenomena need not involve infinite limit systems. I develop my argument by introducing what I regard as the explanatorily relevant property in RG (...) explanations: linearization* property; I then motivate and prove a proposition about the linearization* property in support of my view. As a result, infinite limit systems in RG explanations are dispensable. (shrink)

Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite (...) sums can be equipped with the rich and original structure of a non-commutative ordered ring, and that Bolzano’s views on the mathematical infinite are, after all, consistent. (shrink)

Abstract / 摘要 Relational Quantum Mechanics (RQM) posits that all physical properties are relative to the observer. However, Riedel (2024) identifies a critical logical challenge: the Unrestricted Iteration Principle (UIP), which leads to an infinite regress of relativization. This paper proposes a resolution by introducing the Universal Field of Intelligence (UFI) as the absolute substrate of information. We argue that while interaction outcomes are perspectival, the UFI functions as the Ultimate Observer or the "One True Realm" (一真法界), providing the (...) foundational phase-alignment mechanism that ensures global coherence. By terminating the iteration at the level of the foundational field, this model reconciles the flexibility of relational facts with the ontological necessity of a unified reality. -/- 關係量子力學 (RQM) 主張所有物理屬性皆相對於觀測者。然而，Riedel (2024) 指出了一個關鍵的邏輯挑戰：不限制疊代原則 (UIP)，這會導致相對化的無限回歸。本論文透過引入「宇宙智能場」 (UFI) 作為資訊的絕對基底，提出了一套解決方案。我們主張，雖然交互作用的結果是觀點性的 (perspectival)，但 UFI 作為「最終觀測者」或「一真法界」，提供了確保全域相干的基礎相位對齊機制。透過在基礎場層面終結這種疊代，本模型調和了關係性事實的靈活性與統一實相在學術本體論上的必要性。 . (shrink)

It is possible that a fair coin tossed infinitely many times will always land heads. So the probability of such a sequence of outcomes should, intuitively, be positive, albeit miniscule: 0 probability ought to be reserved for impossible events. And, furthermore, since the tosses are independent and the probability of heads (and tails) on a single toss is half, all sequences are equiprobable. But Williamson has adduced an argument that purports to show that our intuitions notwithstanding, the probability of an (...) infinite sequence is 0. In this paper, I rebut his argument.No Abstract. (shrink)

Beneficence—the part of morality concerned with promoting people's well-being—is widely thought to be both agent-neutral and impartial: it prescribes a common aim to all, and does not favor some individuals over others. This paper explores a problem for agent-neutral, impartial beneficence from the perspective of "individualistic ethics" in the tradition of Harsanyi. The problem reveals that if we want only what is best for each of infinitely many individuals, and we are rational, then we must care about some individuals more (...) than others. I conclude that, on the individualistic approach, value must be fundamentally agent-relative. (shrink)

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)

Physicalism is frequently understood as the thesis that everything depends upon a fundamental physical level. This standard formulation of physicalism has a rarely noted and arguably unacceptable consequence—it makes physicalism come out false in worlds which have no fundamental level, for instance worlds containing things which can infinitely decompose into smaller and smaller parts. If physicalism is false, it should not be for this reason. Thus far, there is only one proposed solution to this problem, and it comes from the (...) so-called via negativa account of physicalism. Via negativa physicalism identifies the physical with the non-mental, such that if everything in the world ultimately depends only on non-mental things, then physicalism is true. To deal with the possibility of worlds without a fundamental level, this account says that physicalism is false in worlds with either a fundamental mental level or an infinite descent of mental levels. Here I argue that there could be a world with an infinite descent of all-mental levels, yet in which physicalism might plausibly be true—thus contradicting the sufficient-for-false condition meant to save physicalism from the threat of infinitely decomposable worlds. This leaves the need for a new dependence-based account of physicalism. (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)

This paper offers a philosophical account of the specific form of romantic love underlying the ideal of love-based marriages. Rather than examining the institution of marriage, it considers marriage as the promise of infinite love between finite persons. Although this promise may seem irrational, even those who never formally marry still invoke phrases like ‘I love you forever’. In three steps, this paper explores what we could possibly mean by infinite love and how it can be rationally promised (...) throughout a finite life. First, I trace the concept of infinite love back to the metaphysical discussions surrounding the emergence of the love-based marriage among German Idealists and Jena Romanticists. Next, drawing on John Searle’s speech act theory, I examine how the ideal of infinite love can be articulated as a promise. Finally, I turn to early existentialist thought—particularly the notions of passion (Lidenskab, Leidenschaft), repetition (Gjentagelsen, Wiederkehr), and the moment (Øjeblik, Augenblick) as developed by Kierkegaard and Nietzsche—to justify the meaning of the marital promise. In short, I propose that instead of interpreting the marital promise as a description of an expected reality, we should approach it as a passionate necessity that discloses the world in a fundamentally indeterminate way. By reframing the marital promise in this light, I aim to show that marital love is compatible both with the ideal of personal autonomy and with an alternative conception of rationality and temporality. (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.

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)

Kant's account of space as an infinite given magnitude in the Critique of Pure Reason is paradoxical, since infinite magnitudes go beyond the limits of possible experience. Michael Friedman's and Charles Parsons's accounts make sense of geometrical construction, but I argue that they do not resolve the paradox. I argue that metaphysical space is based on the ability of the subject to generate distinctly oriented spatial magnitudes of invariant scalar quantity through translation or rotation. The set of determinately (...) oriented, constructed geometrical spaces is a proper subset of metaphysical space, thus, metaphysical space is infinite. Kant's paradoxical doctrine of metaphysical space is necessary to reconcile his empiricism with his transcendental idealism. (shrink)

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)

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.

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)

In his commentary on Aristotle’s _De interpretatione_, Ammonius puts forward an argument for the priority of categorical over hypothetical syllogisms. The argument relies on two of the Five Modes of Agrippa, the modes from infinite regress and from hypothesis. Much of the argument, however, remains unclear and open to doubt. The present chapter sheds new light on the argument by considering it against the backdrop of two related arguments given by Pseudo-Ammonius and Alexander of Aphrodisias in their commentaries on (...) the _Prior Analytics_. The chapter argues that all three arguments originate in Theophrastus’ discussion of Aristotle’s treatment of syllogisms from a hypothesis. They rely on the view that stating a hypothetical proposition _If P, then Q_ does not amount to the unqualified assertion of a conditional proposition, but rather to a conditional assertion of _Q_ on the supposition that _P_. (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)

We present an abstract social aggregation theorem. Society, and each individual, has a preorder that may be interpreted as expressing values or beliefs. The preorders are allowed to violate both completeness and continuity, and the population is allowed to be infinite. The preorders are only assumed to be represented by functions with values in partially ordered vector spaces, and whose product has convex range. This includes all preorders that satisfy strong independence. Any Pareto indifferent social preorder is then shown (...) to be represented by a linear transformation of the representations of the individual preorders. Further Pareto conditions on the social preorder correspond to positivity conditions on the transformation. When all the Pareto conditions hold and the population is finite, the social preorder is represented by a sum of individual preorder representations. We provide two applications. The first yields an extremely general version of Harsanyi's social aggregation theorem. The second generalizes a classic result about linear opinion pooling. (shrink)

This paper is Part II of a trilogy extending earlier works on the ICB ( e.g. 'AGI is Impossible'(Schlereth, 2025)), expanding upon specific domains of the original proof. -/- Building upon the conceptual diagnosis presented in Part I (“The Canary in the Algorith- mic Coal Mine”), this paper provides the rigorous mathematical proof of the Infinite Choice Barrier. (ICB)1 We demonstrate that the limitations of algorithmic cognition described in the prelude are not merely engineering bottlenecks, but formal inevitabilities derived (...) from a triangulation of three fundamental mathematical domains. Specifically, we prove: (1) via Computability Theory (Rice’s Theorem), that semantic frame adequacy is undecidable from within a system; (2) via Information Theory, that entropy diverges in heavy- tailed decision spaces (α≤1), rendering probabilistic inference structurally unstable; and (3) via Algorithmic Complexity (Chaitin’s Incompleteness), that frame-transcendent insights are algorithmically unrecognizable. Furthermore, weunifytheseresultsthroughacategoricalproofusingSheafTheory, demon- strating that “Gluing Failures” between consistent local semantic sections are mathematically necessary in irreducibly infinite decision spaces. -/- ───────────────────────────────────────────── Stack context (added 2026-05-15): -/- • ICB I — Conceptual diagnosis. Schlereth (2025), PhilPapers ID SCHTIC-15. • ICB II — Formal proof, triangulation, Coq appendix. Schlereth (2025), PhilPapers ID SCHTIC-16. • ICB III — Probabilistic case (P-BOSS, hallucination). Schlereth (2025), PhilPapers ID SCHTIC-17. • ICB IV — CSI architecture, General Cognitive Equation. Schlereth (2026), PhilPapers ID SCHAIN-8. • PTAC v1.0 — Coq machine-verified formalization. Schlereth (2026), Zenodo DOI 10.5281/zenodo.20195456 (GitHub repo: z7xt8zzjb5-prog/PTAC_v.1). ─────────────────────────────────────────────. (shrink)

There is a long-standing debate in epistemology on the structure of justification. Some recent work in formal epistemology promises to shed some new light on that debate. I have in mind here some recent work by David Atkinson and Jeanne Peijnenburg, hereafter “A&P”, on infinite regresses of probabilistic support. A&P show that there are probability distributions defined over an infinite set of propositions {\ such that \ is probabilistically supported by \ for all i and \ has a (...) high probability. Let this result be “APR”. A&P oftentimes write as though they believe that APR runs counter to foundationalism. This makes sense, since there is some prima facie plausibility in the idea that APR runs counter to foundationalism, and since some prominent foundationalists argue for theses inconsistent with APR. I argue, though, that in fact APR does not run counter to foundationalism. I further argue that there is a place in foundationalism for infinite regresses of probabilistic support. (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)

The aim of this paper is to examine the medieval posterity of the Aristotelian and Pyrrhonian treatments of the infinite regress argument. We show that there are some possible Pyrrhonian elements in Autrecourt's epistemology when he argues that the truth of our principles is merely hypothetical. By contrast, Buridan's criticisms of Autrecourt rely heavily on Aristotelian material. Both exemplify a use of scepticism.

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)

In this paper we deepen Mundici's analysis on reducibility of the decision problem from infinite-valued ukasiewicz logic to a suitable m-valued ukasiewicz logic m, where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in if and only if it is also valid in m. We also reduce the notion of logical consequence in to the same (...) notion in a suitable finite set of finite-valued ukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued ukasiewicz logic. (shrink)

In PV 3.440ab and 473cd–474ab, Dharmakīrti raises the argument of infinite regress twice. The argument originates from the same argument stated by Dignāga in his Pramāṇasamuccaya 1.12ab1, in which the fault of infinite regress is called aniṣṭhā. In Pramāṇasamuccayavṛtti 1.12b2, Dignāga presents another type of argument of infinite regress driven by memory, which is elucidated by Dharmakīrtian commentators. The arguments were criticized by Kumārila Bhaṭṭa and Bhaṭṭa Jayanta and even more intensively so by two modern scholars, Jonardon (...) Ganeri and Birgit Kellner. In this paper, I first examine the source of the arguments—Pramāṇasamuccayavṛtti 1.12 and its translation, based on which I provide my interpretation of the two models of arguments of infinite regress. I then offer my response, according to Dharmakīrti and his commentators, to Ganeri’s and Kellner’s critiques. By doing so, I attempt to identify the essence of these arguments is and find out to what extent one can defend the infinite regress argument in Dignāga’s and Dharmakīrti’s 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)

This article offers a limited defense of metaphysical “infinitism,” the view that there are, or might be, infinite chains of ontological dependence. According to a widespread presupposition, there must be an ultimate ground of being—most likely, a plurality of fundamental atoms. Contrary to this view, this article shows that metaphysical infinitism is internally coherent. In particular, a parallel with the debate concerning infinitism about epistemic justification is suggested, and an “emergence model” of being is put forward. According to the (...) emergence model, the being of any given entity gradually arises out of an infinite series of progressively less dependent entities—it is not wholly transmitted, as it were, from a basic, ungrounded level to all the dependent ones in a step-by-step fashion. Some objections are considered and rebutted. (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)

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)

An infinite future thus threatens to paralyze utilitarianism. Utilitarians need principled ways to determine which possible infinite futures are better or worse. In this article, I discuss a recent suggestion of Peter Vallentyne and Shelly Kagan. I conclude that the best way forward for utilitarians is, in fact, to by-pass the infinite utility debate altogether. (edited).

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.

Spinoza's Mediate Infinite Mode TAD M. SCHMALTZ IN PART I of the Ethics, Spinoza argued that a modification is infinite just in case it either "follows from the absolute nature of any attribute of God" or "follows from some attribute of God, as it is modified by such a modification" that is infinite. 1 The main purpose of this argument is to bolster the claim later in this text that a finite modification can follow from a divine (...) attribute only insofar as that attribute is modified by another finite modification. 2 Thus it is understandable that in the section that contains the argument Spinoza did not actually affirm the existence of the two kinds of infinite modifications he mentioned, which following standard practice I call "immediate infinite modes" and "mediate infinite modes," respectively.3 Yet in this section he did 'E IP21,D-22,D, G II 65-66/Curley 429-3 o. In the text and notes of this paper I use the following abbreviations pertaining to Spinoza's writings: E: Ethics ; Ep.: Letters ; RDPP: Descartes' Principles of Philosophy ; CM: Appendix containing Metaphysical Thoughts ; KV: Short Treatise ; TdlE: Treatise on the Emendation of the Intellect ; G: C. Gebhardt, ed., Spinoza Opera, 4 vols. , cited by volume and page; Curley: E. Curley, trans, and ed., The Collected Works of.. (shrink)

Klein’s account of epistemic justification, infinitism, supplies a novel solution to the regress problem. We argue that concentrating on the normative aspect of justification exposes a number of unpalatable consequences for infinitism, all of which warrant rejecting the position. As an intermediary step, we develop a stronger version of the ‘finite minds’ objection.

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

Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operations. In this paper, we examine the ITRMs introduced by the third and fourth author (Koepke and Miller in Logic and Theory of Algorithms LNCS, pp. 306–315, 2008), where a register content at a limit time is (...) set to the lim inf of previous register contents if that limit is finite; otherwise the register is reset to 0. The theory of these machines has several similarities to the infinite time Turing machines (ITTMs) of Hamkins and Lewis. The machines can decide all ${\Pi^1_1}$ sets, yet are strictly weaker than ITTMs. As in the ITTM situation, we introduce a notion of ITRM-clockable ordinals corresponding to the running times of computations. These form a transitive initial segment of the ordinals. Furthermore we prove a Lost Melody theorem: there is a real r such that there is a program P that halts on the empty input for all oracle contents and outputs 1 iff the oracle number is r, but no program can decide for every natural number n whether or not ${n \in r}$ with the empty oracle. In an earlier paper, the third author considered another type of machines where registers were not reset at infinite lim inf’s and he called them infinite time register machines. Because the resetting machines correspond much better to ITTMs we hold that in future the resetting register machines should be called ITRMs. (shrink)

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)

Wolff’s relation to Leibniz and Kant’s relation to both are notoriously vexed questions. First, this paper argues that Wolff’s most serious departure from Leibniz consists in his (so far overlooked) rejection of the latter’s infinitism. Second, it contends that the controversies that surrounded Wolff’s early acceptance of infinite causal regress and prompted his conversion to finitism played a prominent role in shaping the theses of Kant’s Antinomies. Whereas Leibniz and the early Wolff considered infinite regress to provide support (...) for the contingency of the world and the existence of God, Wolff’s enemies condemned it as Spinozistic. After Wolff, the claim that an infinite chain of causes is impossible became the standard view among both Wolffian and anti-Wolffian metaphysicians. (shrink)

I have argued for a picture of decision theory centred on the principle of Rationally Negligible Probabilities. Isaacs argues against this picture on the grounds that it has an untenable implication. I first examine whether my view really has this implication; this involves a discussion of the legitimacy or otherwise of infinite decisions. I then examine whether the implication is really undesirable and conclude that it is not.