In some games, the products of the probabilities times the payouts result in a series that is conditionally convergent, which means that the sum can vary based on the order in which the products are summed. The purpose of this paper is to address the question: How should such games be valued? We first show that, contrary to widespread belief, summing in the order determined by the mechanism of the game does not lead to the correct value. We then consider (...) other attempts to value such games. We suggest that games with conditionally convergent series are properly valued by summing the products in increasing order of the absolute values of the game’s payouts. (shrink)

In this reply to Tim Bayne and Yujin Nagasawa, I defend the possibility of a maximal-excellence account of the grounding of the obligation to worship God. I do not offer my own account of the obligation to worship God; rather I argue that the major criticism fails. Thus maximal-excellence can ground an obligation to worship God.

The purpose of this note is to contrast a Cantorian outlook with a non-Cantorian one and to present a picture that provides support for the latter. In particular, I suggest that: i) infinite hyperreal numbers are the (actual, determined) infinite numbers, ii) ω is merely potentially infinite, and iii) infinitesimals should not be used in the di Finetti lottery. Though most Cantorians will likely maintain a Cantorian outlook, the picture is meant to motivate the obvious nature of the non-Cantorian outlook.

In this paper, I present a puzzle involving special relativity and the random selection of real numbers. In a manner to be specified, darts thrown later hit reals further into a fixed well-ordering than darts thrown earlier. Special relativity is then invoked to create a puzzle. I consider four ways of responding to this puzzle which, I suggest, fail. I then propose a resolution to the puzzle, which relies on the distinction between the potential infinite and the actual infinite. I (...) suggest that certain structures, such as a well-ordering of the reals, or the natural numbers, are examples of the potential infinite, whereas infinite integers in a nonstandard model of arithmetic are examples of the actual infinite. (shrink)

In ’The Coherence of Theism’ Richard Swinburne writes that a person cannot be omniscient and perfectly free. In ’The Existence of God’ Swinburne writes that God is a person who is omniscient and perfectly free. There is a straightforward reason why the two passages are not in tension, but recognition of this reason raises a problem for Swinburne’s argument in ’The Existence of God’ (the conclusion of which is that God likely exists). In this paper I present the problem for (...) Swinburne’s argument. I then consider two potential responses and suggest that neither succeeds. (shrink)

Richard Swinburne’s argument in The Existence of God discusses many probabilities, ultimately concluding that God probably exists. Swinburne gives exact values to almost none of these probabilities. I attempted to assign values to the probabilities that met that weak condition that they could be correct. In this paper, I first present a brief outline of Swinburne’s argument in The Existence of God. I then present the problems I encountered in Swinburne’s argument, specifically problems that interfered with my attempt to arrive (...) at values for the probabilities discussed by Swinburne. Finally, I suggest that Swinburne’s argument would be more persuasive if he assigned exact values to his probabilities. (shrink)

In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to be counted (...) by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers. (shrink)

Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In particular, I argue that (...) the infinite numbers should be as much like the finite numbers as possible. Results: Using finite numbers as our guide to the infinite numbers, it follows that infinite numbers are of the structure w + (w* + w) a + w*. This same structure also arises when a large finite number is under investigation. Implications: A first implication of the paper is that infinite numbers may be large finite numbers that have not been investigated fully. A second implication is that there is no number of finite numbers. Third, a number of paradoxes of the infinite are resolved. One change that should occur as a result of these findings is that “infinitely many” should refer to structures of the form w + (w* + w) a + w*; in contrast, there are “indefinitely many” natural numbers. Constructivist content: The constructivist perspective of the paper is a form of strict finitism. (shrink)

Swinburne relies on principle P in The Existence of God to argue that God is simple and thus likely to exist. In this paper, I argue that Swinburne does not support P. In particular, his arguments from mathematical simplicity and scientists’ preferences both fail. Given the central role P plays in Swinburne’s overall argument in The Existence of God , I conclude that Swinburne should further support P if his argument that God likely exists is to be persuasive.

In ‘Infinitism Regained’, Jeanne Peijnenburg argues for a version of infinitism wherein ‘beliefs may be justified by an infinite chain of reasons that can be actually completed’. I argue that Peijnenburg has not successfully argued for this claim, but rather has shown that certain infinite series can be computed.

A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite numbers. I will argue that that this second conception of (...) infinite number is the correct one, and analyze what this means for multiverses. (shrink)

In this article, I argue that it is impossible to complete infinitely many tasks in a finite time. A key premise in my argument is that the only way to get to 0 tasks remaining is from 1 task remaining, when tasks are done 1-by-1. I suggest that the only way to deny this premise is by begging the question, that is, by assuming that supertasks are possible. I go on to present one reason why this conclusion (that supertasks are (...) impossible) is important, namely that it implies a new verdict on a decision puzzle propounded by Jeffrey Barrett and Frank Arntzenius. (shrink)

In a recent paper, Sylvia Wenmackers and Leon Horsten discuss how the concept of a fair infinite lottery can best be extended to denumerably infinite lotteries. Their paper uses and builds on the alpha-theory of Vieri Benci and Mauro Di Nasso. The purpose of this paper is to demonstrate a potential problem for alpha-theory.

When two omnipotent beings are randomly and sequentially selecting positive integers, the being who selects second is almost certain to select a larger number. I then use the relativity of simultaneity to create a paradox by having omnipotent beings select positive integers in different orders for different observers.

In this paper, two concepts of completing an infinite number of tasks are considered. After discussing supertasks, equisupertasks are introduced. I suggest that equisupertasks are logically possible.

In this brief reply to Stephen Davis, I argue that Davis’s separationist position, wherein those who remain eternally apart from God do so by choice, is internally contradictory in that it leads to universalism.

In “How an Unsurpassable Being can Create a Surpassable World,” Daniel and Frances Howard-Snyder employ a fascinating thought experiment in anattempt to show that a morally unsurpassable being can create a surpassable world. Imagine that for each positive integer there is a world that a good,omnipotent, omniscient being can create. Jove randomly selects a number and creates the corresponding world; Thor simply creates world 888. The Howard-Snyders argue that it is logically possible that Jove is morally unsurpassable. William Rowe counters (...) that Thor morally surpasses Jove, thus contradictingthe claim that Jove is morally unsurpassable. Does either Jove or Thor morally surpass the other? How do their strategies compare? Could a morally unsurpassable being employ Jove’s strategy? The purpose of this paper is to answer these questions. (shrink)

The purpose of this paper is to suggest that we are in the midst of a Cantorian bubble, just as, for example, there was a dot com bubble in the late 1990’s.

The purpose of this paper is to point the way to the money tree. Currently, almost all investment professionals think that outperformance requires an “edge,” that is, the ability to predict the future to some degree. In this paper, I suggest that money can be made in a 0, or even slightly negative, expected value environment by carefully choosing investment/bet sizes. Philosophical considerations are found mainly in Section 4.

The purpose of this paper is to dissolve paradoxes of the infinite by correctly identifying the infinite natural numbers.

Consider a subset, S, of the positive integers. What is the probability of selecting a number in S, assuming that each positive integer has an equal chance of selection? The purpose of this short paper is to provide an answer to this question. I also suggest that the answer allows us to determine the relative sizes of two subsets of the positive integers.

In this brief paper, I present a puzzle for consideration.

In ‘The Train Paradox’, I argued that sequential random selections from the natural numbers would grow through time. I used this claim to present a paradox. In response to this proposed paradox, Jon Pérez Laraudogoitia has argued that random selections from the natural numbers do not grow through time. In this paper, I defend and expand on the argument that random selections from the natural numbers grow through time. I also situate this growth of random selections in the context of (...) my overall work on infinite number, which involves two main claims: 1) infinite numbers, properly understood, are the infinite natural numbers in a nonstandard model of the reals, and 2) ω is potentially infinite. (shrink)

Recently Timothy Williamson asked ‘How probable is an infinite sequence of heads?’ In this paper, I suggest the probability of an infinite sequence of heads.

It is commonly believed that when a finite value is received in a game that has an infinite expected value, it is in one’s interest to redo the game. We have argued against this belief, at least in the repeated St Petersburg two-envelope case. We also show a case where repeatedly opting for a higher expected value leads to a worse outcome.

In this reply to Jesse Steinberg’s ‘God and the possibility of random creation’, I suggest a procedure whereby a being such as God could randomly select a number from an infinite set.

In "Religion and Scientific Method," George Schlesinger presented a strikingly original theodicy. In this paper, I explain the strategy underlying Schlesinger's argument. I then present a parallel argument to indicate the weakness of Schlesinger's theodicy. Finally, I show that Schlesinger's theodicy assumes a false principle, and therefore fails.

Swinburne relies on principle P in ’The Existence of God’ to argue that God is simple and thus likely to exist. In this paper, I argue that Swinburne does not support P. In particular, his arguments from mathematical simplicity and scientists’ preferences both fail. Given the central role P plays in Swinburne’s overall argument in ’The Existence of God’, I conclude that Swinburne should further support P if his argument that God likely exists is to be persuasive.

Steven M. Cahn presented a fascinating and recalcitrant puzzle. Imagine that you own stock in a company that has undertaken an immoral course of action. Most people believe that selling the stock is the right thing to do. But if you sell the stock, you thereby abet the buyer in an immoral course of action, namely the buyer’s coming to own tainted stock. Is principled divestiture possible? If so, what are the moral grounds on which principled divestiture stands? These questions (...) have proven difficult. Cahn answered many attempted replies to his puzzle. In this journal, Saul Smilansky also attempted to answer the divestiture puzzle. Smilansky concluded, “Attempting to divest by offering the stock for sale is, however, morally problematic. It is problematic because it involves the bad faith of putting something forward as morally acceptable while claiming it to be morally tainted.” Smilansky continued, “One needs to get rid of the stock and take it off the market at the same time, i.e., make it impossible for others to purchase the stock.” Smilansky thereby implies that one should destroy the stock. However, Cahn had already replied to this move, noting that destroying one’s stock would increase the moral culpability of other owners. (shrink)