Grim Reaper Paradoxes and Causal Finitism

The previous article was about how evidence bears on a hypothesis. This one is about a different sort of argument altogether, one that tries to rule out a possibility rather than weigh it, and that in its best contemporary form is genuinely hard to answer.

The Kalam cosmological argument needs a premise saying that the universe began to exist. Empirical cosmology supplies part of the case for that premise and the philosophical arguments against an infinite past supply the rest. The intermediate tier gave those arguments in their traditional form. They have since been rebuilt, and the rebuilt versions do not depend on the traditional ones being any good, which is fortunate, because the traditional ones are weak.

The two traditional arguments

The hotel is a standing fixture of the subject and is usually attributed to a lecture David Hilbert gave in the 1920s. It reached a wide readership through George Gamow's One Two Three... Infinity, published in 1947, and entered philosophy of religion largely through William Lane Craig's The Kalām Cosmological Argument, Macmillan (1979), whose first part traces the medieval sources of the argument in al-Kindi, Saadia, and al-Ghazali and whose second part defends a version of it.

The illustration is doing work for a distinction the tradition drew long before Cantor. A potential infinite is a collection that can always be extended and is never complete, as the series of counting numbers is when it is thought of as something we go on generating. An actual infinite is a completed totality with infinitely many members all at once. Aristotle allowed the first and rejected the second, and the medieval arguments the tier has met are attempts to rule out the second in the world while leaving mathematics alone. The hotel is meant to show what accepting an actual infinite would commit you to.

Imagine a hotel with a room for every counting number, all of them occupied. A new guest arrives. There is no free room, yet the manager accommodates them: everyone moves along one room, the guest in room 1 goes to room 2 and so on forever, and room 1 falls vacant. The hotel was full and it took another guest without turning anyone away. Now infinitely many new guests arrive. Everyone moves to the room with twice their current number, all the odd rooms fall vacant, and infinitely many arrivals are housed in a hotel that was already full.

Subtraction is where the argument thinks it has you. Suppose every guest in an odd-numbered room checks out. Infinitely many guests have left and infinitely many remain, so the hotel is as full as before. Now instead suppose that every guest from room 4 onwards checks out. Infinitely many guests have left, exactly as in the first case, and three remain. The same operation, performed on the same collection, removing the same number of guests, leaves different numbers behind. On the face of it that is not strange but incoherent, and the argument concludes that no such collection can exist in reality.

The standard reply is decisive against the argument as stated, and worth getting right. Nothing in the story is a contradiction. It is a set of consequences of the cardinality of a countably infinite set, and Cantor's theory of transfinite numbers, in which they are theorems, is consistent. Addition and multiplication are perfectly well defined for infinite cardinals and they behave exactly as the hotel describes. Subtraction and division are not defined, and the two check-out scenarios are why: an operation that would have to yield different answers for the same inputs is not an operation. Undefined is not the same as contradictory, in just the way that division by zero is undefined without making arithmetic incoherent. Graham Oppy develops this line at length in Philosophical Perspectives on Infinity, Cambridge University Press (2006).

There is a reply available to the defender, and it is a retreat. The claim, properly stated, is not that transfinite arithmetic is inconsistent but that a consistent formal structure need not be capable of physical instantiation. That is true, and it changes the argument's status entirely. It is no longer a proof of anything. It is an appeal to an intuition about which structures could be real, made in the one area where intuition has the worst track record in the history of thought.

The second traditional argument does not concern whether an actual infinite could exist. It concerns whether a particular one could be arrived at.

The first premise is supported by observing that no amount of adding one to a finite collection produces an infinite one, and by the impossibility of counting to infinity. Whichever stage you inspect, the collection is finite and the next addition leaves it finite, so at no stage does the total become infinite.

The reply is that the support smuggles in a starting point that the hypothesis denies. Counting to infinity is impossible because counting starts somewhere. A beginningless past has no first event, so there is no stage at which the series was finite and had the job of growing. Every past event has only finitely many events between itself and now, which is everything that needs to be true for each of them to have been reached, and no event ever had to traverse infinitely many predecessors, because each event's predecessors were themselves already in the past. The argument's picture is of a series being built from a beginning. Its target is a series with no beginning, and the picture does not fit.

Neither argument has persuaded anyone not already persuaded. That is not a rhetorical dismissal, it is the reason the literature moved on, and the direction it moved in is worth following closely.

Supertasks

The term is James Thomson's, from "Tasks and super-tasks", Analysis 15(1): 1-13 (1954), and the modern discussion begins there rather than with Zeno, though Zeno's dichotomy is the ancestor.

Thomson's example is a lamp. Its switch is flipped at a moment, then again after half the remaining interval, then after half of what remains again, and so on without end, so that infinitely many flips are completed inside a finite time. Ask what state the lamp is in at the end. It cannot be on, since every flip that turned it on was followed by one that turned it off. It cannot be off, for the same reason in reverse. Thomson took this to show that the supertask is impossible.

Paul Benacerraf's reply, in "Tasks, Super-Tasks, and the Modern Eleatics", The Journal of Philosophy 59(24): 765-784 (1962), is the standard one and it is correct. The description of the lamp specifies its state at each moment strictly inside the interval. It says nothing whatever about the state at the limit, because the limit is not one of the moments described. There is no contradiction, only an underspecified story, and the reader has mistaken the absence of a stipulation for an impossibility. Any state at the end is consistent with everything in the setup.

It is worth noticing why the familiar treatment of Zeno does not settle Thomson's case. Zeno's dichotomy is answered by observing that an infinite series of intervals can sum to a finite total, so that traversing infinitely many sub-distances in a finite time involves no arithmetical difficulty. Thomson granted all of that. His lamp flips are already packed into a finite total by construction, and the question he raised was not whether the series converges but what state the object is in once it has. Convergence of the times says nothing about the value of a quantity that oscillates and has no limit, which is exactly the situation the lamp is in.

A second supertask sharpens the point in a different direction. Suppose that at each of infinitely many steps within a finite interval, ten balls are added to a vase and one is removed. If the balls are numbered and the one removed is always the lowest-numbered remaining, then every ball is removed at some step, and the vase ends up empty despite the number of balls growing without bound at every stage. If instead the ball removed is always the highest-numbered, infinitely many remain. Same additions, same number of removals at every step, different endings, and the difference is fixed entirely by which balls were labelled how. Here the setup does determine the outcome, and the outcome is merely strange rather than contradictory, which is a useful calibration for what follows.

That result matters, and it cuts in an unobvious direction. Benacerraf did not show that supertasks are possible. He showed that the lamp fails to show they are not, which left the question open and left a template for a better argument. A better argument would have to produce a case where the setup does determine the outcome, and determines it inconsistently.

The Benardete construction

The family takes its name from José Benardete's Infinity: An Essay in Metaphysics, Oxford University Press (1964), which contains several. In one of them a traveller sets out to walk a mile and a god stands ready to raise an impassable wall at the half-mile mark if the traveller reaches it, another ready to raise one at the quarter-mile mark on the same condition, and so on inward without end. No wall is ever raised, because before any given wall's mark is reached an earlier one would have gone up. Yet the traveller cannot advance at all, since advancing any distance means passing infinitely many marks at which a wall would have been raised. The traveller is stopped, and stopped by nothing.

The structural ingredients are worth naming, because every case in the family has all of them and dropping any one dissolves the paradox. There are infinitely many agents. Each has a conditional instruction whose trigger is that no earlier agent has acted. The agents are ordered so that there is no earliest. And the whole arrangement fits inside a finite region of space or time.

Watch what happens when each ingredient is removed, since this is the quickest way to see what the case is really about. Give the series a first member and everything settles: the first agent acts, every later one finds the condition already met and does nothing, and the story is dull rather than contradictory. Make the instructions unconditional and the story is again dull, since all of them act and the question of who acted first has an answer even if that answer is that infinitely many acted at once. Spread the agents out so that their times have no upper bound and the contradiction lifts, because now for each agent there is an earliest one to consider. The paradox lives entirely in the combination of a conditional trigger with an ordering that has no least element, packed into a bounded interval, and any account of what has gone wrong has to say something about that combination rather than about infinity in general.

The version that has driven the recent literature runs like this. Fred is alive at noon. Reaper 1 is set to act at one o'clock, reaper 2 at half past twelve, reaper 3 at quarter past, and so on, each reaper's time half as far after noon as the last, so that the appointed times pile up toward noon without ever reaching it. Every reaper follows the same instruction: on waking, check whether Fred is already dead; if he is, do nothing at all; if he is not, kill him instantly.

Fred is certainly dead by one o'clock. If he were alive at one, reaper 1 would have killed him, and if reaper 1 found him dead then someone earlier had done it. So a killing occurred.

But no reaper can have been the one who did it. Take any reaper you like, say reaper 400. If reaper 400 killed Fred, then Fred was alive when reaper 400 woke. That means reaper 401, whose time is earlier, found Fred alive and killed him, which contradicts Fred being alive later. The same reasoning defeats every candidate, because for every reaper there is an earlier one, and there is no first reaper for the argument to bottom out at.

So Fred dies and nobody kills him. Each individual reaper's instruction is unobjectionable. The set is inconsistent.

The argument that concerns philosophy of religion takes that inconsistency and asks what has to be given up.

Robert Koons develops the argument into a version of the Kalam in "A New Kalam Argument: Revenge of the Grim Reaper", Noûs 48 (2014), pp. 256-267. Alexander Pruss gives the most sustained treatment of the whole family in Infinity, Causation, and Paradox, Oxford University Press (2018), and his conclusion is the one the tier needs next.

Causal finitism

Pruss's strategy is what gives the thesis its interest, and it is unlike most arguments in philosophy of religion. Each Benardete case can be resolved on its own, and each such resolution is a piece of special pleading fitted to that case. The lamp can be handled by Benacerraf's point about limits. The walls can be handled by denying that the gods could all be so disposed. The reapers can be handled by insisting that some reaper must fail. What none of these does is explain why a new paradox appears whenever the same structure is rebuilt with different furniture. Pruss's proposal is that one principle kills the whole family at once, which is the sort of consideration that supports a thesis in any other part of philosophy.

Causal finitism does kill them. Every case in the family requires an event, Fred's death or the traveller's being stopped, with infinitely many causal antecedents, since infinitely many agents' dispositions bear on it. Rule that out and no case in the family can be set up. The thesis is also compatible with a great deal that its neighbours are not: it does not require rejecting the actual infinite in mathematics, does not require that there be finitely many things, and does not by itself require that space or time be discrete.

The route to theism is short from there. If nothing can have infinitely many causal antecedents, then a causal series per se cannot be infinite and neither can a beginningless series of past events, since either would give present events infinitely many antecedents. The universe therefore has a first cause. The cosmological arguments of the intermediate tier reach their terminus by an argument that never mentions God, never mentions explanation, and never depends on the principle of sufficient reason.

What the paradox actually establishes

Here is where care is needed, because the argument's supporters and its critics often agree about the paradox and disagree about what follows, and the disagreement is not the one they think they are having.

The Grim Reaper case shows that some assumption in the setup is false. It does not show which. Causal finitism is one candidate. It is not the only one, and the others are not obviously worse.

Time might be discrete rather than continuous. If there is a shortest interval, the reapers cannot be packed into ever-smaller gaps, there is a first reaper, and the contradiction never arises. This resolution costs a substantial commitment in the philosophy of physics and buys nothing for theism, which is presumably why it is less popular in this literature than its merits warrant.

The conditional instructions might be jointly unsatisfiable in a way that makes the setup describe nothing. On this diagnosis the case is a trick of specification rather than a discovery about causation. Compare a story about a barber who shaves exactly those who do not shave themselves: the story is inconsistent, and the right response is that there is no such barber, not that we have learned something about shaving. The reaper case is more interesting than the barber because its inconsistency is not visible in any one instruction, only in the infinite collection. Whether that difference is deep or merely a matter of how hard the inconsistency is to spot is exactly the contested question, and it has not been settled.

Or the possibility premise might fail for the mundane reason that infinitely many agents cannot be arranged this way in the actual world. That would leave the paradox with no bearing on what is metaphysically possible, and so none on whether the past could be infinite.

A fourth option is worth separating from the third because it is often run together with it. One might hold that the reapers' instructions are individually fine and jointly satisfiable, and that what fails is the assumption that the collection determines a unique outcome. This is Benacerraf's point about the lamp, transposed. If the setup simply does not fix what happens, then there is no contradiction to explain, only a story that stops short of describing a complete situation. The reason this is harder to sustain for the reapers than for the lamp is that the reaper case appears to fix the outcome twice over, once by the argument that Fred must be dead and once by the argument that no reaper killed him, and both arguments use only the stated instructions. Whether they really do is a technical question about how the conditionals should be regimented, and there is no consensus on the answer.

There is also a step at the end that this literature tends to hurry past, and it should not be hurried past here. Suppose causal finitism is established and the past is therefore finite. What follows is that there is a first cause, which is where the intermediate tier's cosmological arguments also arrived, and which is a long way short of God. Nothing in the Grim Reaper case bears on whether that first cause is personal, powerful, good, or one rather than many. The argument's contribution is to a single premise of a much longer chain, and the remaining links have to be supplied by arguments of an entirely different kind, most of which the intermediate tier already found contested.

Next

The arguments of this article and the last both aim at conclusions about what must exist by reasoning about what is possible. The next article takes that method to its purest case, the argument that runs from the bare possibility of a maximally great being to its actual existence, and examines the parallel arguments that appear to run just as well to conclusions nobody accepts.