By Daniel Rubio

The combination of standard Christian doctrines and standard theories of practical reason, especially those in which expected utility is important, creates a series of problems. Problems stemming from infinity. My goal is to make the problems clear so that even those without much training in mathematics can appreciate and think about them. I’ll begin with an informal explanation of some of the unique mathematical features of infinite numbers, and then I will show how this combines with standard Christian teachings to make standard theories of practical reason misfire. I will indulge some oversimplifications in doing this, hopefully in exchange for greater clarity and accessibility. The second part of the post will deal with potential solutions to these problems.

Infinity in mathematics

Suppose we want an answer to the question, “What is the sum of the positive integers?” We can formulate the equation using simple concepts: 1 + 2 + 3 + 4 +…. Yet, there is no answer among our familiar numbers. Any real number (roughly: any number with a decimal representation) we chose is too small. So we must leave the question unanswered, or add a new number. Since the question seems somewhat sensibly posed, the simplest solution is to add a new number to the real line that is greater than any real number (and consequently any integer). The symbol ∞ is often used for this number, and the new system formed by adding it and its negative mirror is called the ‘extended reals.’

This is not the only sum we might be interested in that has infinitely many terms. A tempting first thought might be that any infinite sum with all positive terms should add to ∞. But this turns out to be false. Consider the sum 1/2 + 1/4 + 1/8 + 1/16… No matter how many terms in this series you sum together, it never goes above 1, and in the limit converges to it. So standard definitions of an infinite sum say that its sum is 1. 

Things stay somewhat intuitive when we are only adding positive numbers. But the addition of negative terms introduces a twist. Consider the series 1 - 1/2 + 1/3 - 1/4 + 1/5… This series does converge, but if we flip all of the minus signs to plus signs (effectively taking the absolute value of each term), it no longer does. Series like this are called conditionally convergent, and they introduce one of the most counterintuitive new properties of infinite summation. 

Finite sums are commutative. 2 + 3 = 3 + 2. It doesn’t matter in what order you sum up the numbers, you always get the same result. Infinite sums are not. Conditionally convergent series can be rearranged to sum to any number (or to diverge to positive or negative infinity). The full proof can be found here, but the technique is somewhat intuitive. Say I want to hit ¾ with my sum. First I separate out my positive and negative terms (each of which by themselves form series that diverge to +/- infinity). Then I line up some positive numbers until I go over ¾. Then I add in negative terms to take me under ¾. Then I add in positive terms to get over ¾. So on and repeat. Because the distance from ¾ will be limited by the last term I used to ‘cross’ it, as we go deeper into the series and so have only smaller and smaller terms available, it will converge to ¾ in the limit. 

Suppose we want to answer the question, “how many even numbers are there?” We have a familiar method for answering how-many questions. We count. Given a collection of things, we count them by matching each member with ever-increasing numbers. In more abstract terms: we count n things by creating a 1:1 correspondence between the things and the numbers between 1 and n. In fact, one common definition for when two collections of things have the same number of items is when there exists a 1:1 correspondence between them. 

Georg Cantor made this idea the centerpiece of his infinitary mathematics (involving roughly what we call the cardinal and ordinal numbers), and it has delivered some surprising results. The first hint of these surprises came when Galileo proved that there is a 1:1 correspdonence between the even (and also the odd) numbers and the natural numbers (for our purposes, think of these as the numbers that go 1,2,3…). Just map each even natural number x to 2x (or 2x - 1 for the odds). Conceptually that’s somewhat bizarre, since every even number is a natural number, but not vice-versa. It pits two important ideas about size against each other: the idea that adding one thing always creates a larger collection, and the idea that a 1:1 correspondence shows that two collections have an equal number of things. 

It turns out that taking either of those ideas as bedrock can lead to a mathematically rigorous theory of the infinite. Cantor chose the idea of 1:1 correspondence as establishing equal number, and built the standard theory of infinite numbers on it. That theory has proven to be incredibly fruitful, but it does inherit some of the conceptual weirdness from its foundational concept. 

The second major surprise came when, despite Galileo’s results about reals, evens, and odds, Cantor proved that there are different sizes of infinite number. The infamous Diagonalization Argument showed that there are more real numbers than there are natural numbers, and more generally given a set, the set of all of its subsets is a larger set. 

A third counterintuitive result came when Cantor formulated the arithmetic for his cardinal numbers. Cardinal numbers have what is known as the Absorption Property: When adding up numbers that include an infinite one, the sum is simply the largest infinity in the equation. The same for multiplication.

How infinities generate problems

There are fundamentally two ways infinity can enter into ethical or decision problems. The first is in the size of various goods or prizes. The classical example of this kind of problem is Pascal’s Wager, first formulated by one of the founders of modern probability theory, Blaise Pascal. In very simplified form: Pascal argued that it is rational to adopt a religious lifestyle, regardless of how likely one thought it was that a god exists, so long as you were not certain of atheism. Pascal thought there were four relevant possibilities: a god exists and you are religious; a god exists and you are not religious; no god exists and you are religious; no god exists and you are not religious. But if a god exists and you are religious, you receive an infinitely valuable afterlife. In any of the other situations, by contrast, you receive a finitely valued life and then nothing. Adding a doctrine of hell to make the ‘god exists & not religious’ situation infinitely disvaluable is an option, but not necessary for the argument. The only needed theology is that there is a god who rewards all and only that god’s worshippers. 

So set up, we can now calculate the expected utilities for each option. And we quickly find that the expected utility of being religious overwhelms the expected utility of not being religious, regardless of what (non-zero, real) numbers we plug in for the probabilities. 

There’s nothing special about the religious context here. Any time we have infinite utilities in play, the absorption property of cardinal arithmetic tells us that whatever course of action has a non-zero chance at delivering the largest infinity has the largest expected utility, and if infinities of the same size could come from multiple choices, those choices have the same expected utility. 

Things get even more complicated if one course of action could result in a positive infinity or a negative infinity of the same size. Generally speaking, subtraction between infinities of the same size is undefined. So in that situation, expected utility theory falls silent. 

Another way infinity can cause trouble in ethical and decision theoretic problems is via an infinite state space. A problem’s state space is roughly the set of possibilities over which the agent is uncertain. Thus even if we make a rule that all utilities have to be finite, we can still run into infinity problems. 

The classical example of this is the St. Petersburg Game. Like Pascal’s wager, it is a problem formulated by some of the earliest decision theorists (Nicholas Bernoulli) that still hasn’t been given a fully satisfactory solution. In the St. Petersburg Game, a coin is flipped until it lands heads. If it lands heads on toss n, the player is paid $2n. Because the possibilities for the first heads are infinite (the natural numbers), the expected value of playing the game is just the sum 1+1+1+1+1…, which diverges to positive infinity,  as the table below shows.  

But of course, any actual play will terminate after finitely many flips, so the game must have a finite payout. So what is a fair price for the game?

The St. Petersburg game is weird, but it is not the most trouble we can make with an infinite state space. By exploiting conditionally convergent series’s, we can create games like Nover and Hájek’s Pasadena Game, whose expected value can sum to whatever number we want depending on the order in which we add up the states. 

The general lesson here is that we can use any weirdness in the mathematics of infinity to create games or choice situations that seem paradoxical and/or present difficult challenges to our theories of rational or moral decision making. Some of these situations can be motivated in ways that Christians/Christian ethics will find very difficult to avoid.

Problems of Infinity in Christian Ethics

Once again, I am going to focus on raising rather than solving problems here. I will discuss three problems for Christian ethics that infinity raises. 

The first problem comes from the afterlife. In the good case, Standard Christian doctrine about the beatific vision places it on another plane compared to finite goods, which is most naturally represented with an infinite number. Likewise, an eternal afterlife will most naturally have an infinite state space. Assuming each day of afterlife is good (and the quality doesn’t decrease too quickly), this will naturally aggregate to an infinite positive number. Similar remarks apply, mutatis mutandis, to the bad case and negative infinities. 

If the afterlife is either infinitely valuable or infinitely disvaluable, this threatens to trivialize decisions in this life. Every option will have one of three possible expected values: positive infinity, negative infinity, or undefined. And this will be regardless of the other goods involved or how the non-zero probabilities stack up. 

A more specific version of this problem applies specifically to acts that have a chance of changing someone’s afterlife from hell to heaven. It will turn out that the deprivation of any finite good or infliction of any finite harm will have a positive or undefined expected utility so long as it has a non-zero chance of flipping someone from hellbound to heavenbound. In a consequentialist or even consequence-sensitive ethic (e.g. threshold deontology), this will make all sorts of things uncomfortably easy to defend. 

A final problem comes not from the infinite goods/harms of the afterlife, but from the infinity associated with God’s goodness. One way of conceiving of God’s goodness is not just as an infinite number (which at least conceptually could be surpassed), but as Ω, Cantor’s largest infinite number. Cantor himself endorsed this view, and wrote to the Vatican explaining his discovery. Recent work in philosophy of religion (most notably from Mark Johnston and Mark Murphy) has taken up this thought. An important consequence of this view is that a world containing God cannot get any better or any worse. Acts may be able to generate more or less value in that world, but because God’s value is already so great, none of this addition or subtraction of value can affect the total value of the world. Consequently, objectives like furthering the good or making the world a better place are misplaced, a problem that will seem especially sharp for effective altruists. 

None of these problems are insoluble, although there is no concensus as to the best solution. For better or worse, the problems infinity raises for practical rationality are often treated as avoidable simply by embracing finitism or dismissed as involving weird or far-fetched scenarios. For those who take the teachings of Christianity seriously, these are not easy options to take. 

Solutions

Now we are going to canvas solutions to these problems. We will look at three families of solutions. In what follows, I will answer the following questions for each family of views: which problems does it handle well, which problems does it handle poorly, and what are its prospects for improvement? I won’t generally speaking dwell on the objections that each of these views faces, but here issue a general caveat that every view I discuss here is controversial. 

Altering the decision rule

The first alters the decision rule. These solutions see expected utility formalism as the root of the problem, and propose to either supplement or entirely replace it with something else. Many of these views end up as conservative extensions of finitistic expected utility theory, agreeing with all expected utility verdicts on cases involving finite state spaces with finite utilities involved. Primary examples include layered decision rules (that is, strategies that introduce a primary decision rule and then employ secondary rules when the primary rule issues tied verdicts or fails to issue a verdict at all) such as in Meacham’s Distance Minimizing Theory and alternative formalisms such as Bartha’s Relative Utility Theory and Colyvan’s Relative Expectation Theory. 

There’s a high degree of flexibility among these views, given their diversity, but they tend to do better with (and often are explicitly designed for) games with infinite state spaces. Layered decision rules can handle games with competing infinite expected values of the same magnitude, e.g. by telling us to break ties between options with the same expected utility by taking the one with a higher probability of getting the infinite prize, but they tend not to generalize well. Since the main problems that result from a combination of expected utility and Christian belief tend not to require infinite state spaces, this family of views tends to be less focused on the problems we are trying to solve. 

Altering the number and arithmetical system

The second family of solutions alters the number and arithmetical systems we use to represent and manipulate probabilities and utilities. These views tend to preserve (either entirely or with only minimal alterations) the principles of rationality (such as the requirement that we not have preference cycles) that typically underpin expected utility theory while using non-standard mathematical tools in their representations. 

These views tend to employ non-standard number systems as the values of probabilities and utilities, and they tend to adopt the arithmetic that comes naturally with those non-standard systems. The two main options are hyperreal numbers, and accompanying hyperreal decision theories such as that of Frederik Herzberg, and surreal numbers and accompanying surreal decision theories such as that of Eddy Keming Chen and Daniel Rubio. Non-standard arithmetic generally sheds the absorption property that plagues cardinal arithmetic, and non-standard numbers systems usually admit of much finer distinctions than the standard cardinal and ordinal numbers. This makes them well-suited to model infinite (dis)values that attach to a single state of the world. Generally speaking, if p and q are probabilities, and N is a non-standard infinite number, then pN ≥ qN if and only if p ≥ q. This means that having infinities on multiple sides of an expected utility calculation no longer leads to triviality or paralysis. It also means that (if their relative sizes align correctly), a sufficiently small chance of an infinite payout can be outweighed by a sufficiently better chance at a finite payout. This is especially helpful with the first two of our problems. However, the relatively underdeveloped state of non-standard calculus means that these views usually face limitations when dealing with infinite state spaces.

Revisions to order-theoretic constraints on rational preference

The third family of views proposes revisions to the order-theoretic constraints on rational preference themselves. Sometimes this can be done in the name of preserving expected utility in infinitistic environments, and sometimes it can be done to keep certain kinds of infinitistic situations from arising. There are two major members of the family: bounded utility views, and probability discounting views. These views tend to have more success in dealing with the problems raised by infinite state spaces than in dealing with single states with infinite (dis)values attached. 

The main idea behind bounded utility views (which may have its earliest expression in the Bernoullian idea of declining marginal utility of money) is that rational preferences don’t allow values to get too high (or to grow too quickly). Jeffrey Russell has shown that when you extend an important standard axiom of finite expected utility theory relating conditional and unconditional preferences (the sure-thing principle, which roughly says if you prefer x to y given z and x to y given not-z, then prefer x to y full stop) to cover countable infinities, you get a kind of bounded utility view. Bounded utilities deal with infinities (or sufficiently large/fast-growing ones) by banning them. Unfortunately, this simply flies into the face of the theological motivations that generate the problems we’re interested in (such as the beatific vision being infinitely better than earthly goods), so bounded utilities are unlikely to provide a satisfactory answer. 

Probability discounting, by contrast, has more promise. The main idea here (also Bernoullian) is that certain non-zero probabilities can rationally be ignored. There are various motivations for this sort of view, with papers by Nicholas Smith, Bradley Monton, and Frank Hong giving a representative sample. This could be especially helpful with the second problem, since it will license excluding certain courses of action that only have a small probability of changing the valence of someone’s afterlife from consideration. Bad news for would-be Torquemadas. It is less helpful with the first and third problems, since they don’t generally depend on small probabilities. 

Reviewing the solutions

Where does this leave us? While no solution or family of solutions is fully adequate, we have at least the beginnings of a handle on the first two problems. We have a number of techniques for keeping infinities from creating paralysis and justifying arbitrarily harmful acts, although none of them is good for all problems. The third problem, on the other hand, isn’t at its core a technical issue. It’s relatively straightforward to give axiological models where the value of God can saturate the universe and crowd out all other values, and to give axiological models where it does not. Whether it actually does is more a question for theologians and philosophers of religion than for theorists of rationality.