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A supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time.[1] Supertasks are called hypertasks when the number of operations becomes uncountably infinite. A hypertask that includes one task for each ordinal number is called an ultratask.[2] The term "supertask" was coined by the philosopher James F. Thomson, who devised Thomson's lamp. The term "hypertask" derives from Clark and Read in their paper of that name.[3]

History

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Zeno

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Motion

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The origin of the interest in supertasks is normally attributed to Zeno of Elea. Zeno claimed that motion was impossible. He argued as follows: suppose our burgeoning "mover", Achilles say, wishes to move from A to B. To achieve this he must traverse half the distance from A to B. To get from the midpoint of AB to B, Achilles must traverse half this distance, and so on and so forth. However many times he performs one of these "traversing" tasks, there is another one left for him to do before he arrives at B. Thus it follows, according to Zeno, that motion (travelling a non-zero distance in finite time) is a supertask. Zeno further argues that supertasks are not possible (how can this sequence be completed if for each traversing there is another one to come?). It follows that motion is impossible.

Zeno's argument takes the following form:

  1. Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
  2. Supertasks are impossible
  3. Therefore, motion is impossible

Most subsequent philosophers reject Zeno's bold conclusion in favor of common sense. Instead, they reverse the argument and take it as a proof by contradiction where the possibility of motion is taken for granted. They accept the possibility of motion and apply modus tollens (contrapositive) to Zeno's argument to reach the conclusion that either motion is not a supertask or not all supertasks are impossible.[citation needed]

Achilles and the tortoise

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Zeno himself also discusses the notion of what he calls "Achilles and the tortoise". Suppose that Achilles is the fastest runner, and moves at a speed of 1 m/s. Achilles chases a tortoise, an animal renowned for being slow, that moves at 0.1 m/s. However, the tortoise starts 0.9 metres ahead. Common sense seems to decree that Achilles will catch up with the tortoise after exactly 1 second, but Zeno argues that this is not the case. He instead suggests that Achilles must inevitably come up to the point where the tortoise has started from, but by the time he has accomplished this, the tortoise will already have moved on to another point. This continues, and every time Achilles reaches the mark where the tortoise was, the tortoise will have reached a new point that Achilles will have to catch up with; while it begins with 0.9 metres, it becomes an additional 0.09 metres, then 0.009 metres, and so on, infinitely. While these distances will grow very small, they will remain finite, while Achilles' chasing of the tortoise will become an unending supertask. Much commentary has been made on this particular paradox; many assert that it finds a loophole in common sense.[4]

Thomson

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Further information: Thomson's lamp

James F. Thomson believed that motion was not a supertask, and he emphatically denied that supertasks are possible. He considered a lamp that may either be on or off. At time t = 0 the lamp is off, and the switch is flipped on at t = 1/2; after that, the switch is flipped after waiting for half the time as before. Thomson asks what is the state at t = 1, when the switch has been flipped infinitely many times. He reasons that it cannot be on because there was never a time when it was not subsequently turned off, and vice versa, and reaches a contradiction. He concludes that supertasks are impossible.[5]

Benacerraf

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Paul Benacerraf believes that supertasks are at least logically possible despite Thomson's apparent contradiction. Benacerraf agrees with Thomson insofar as that the experiment he outlined does not determine the state of the lamp at t = 1. However he disagrees with Thomson that he can derive a contradiction from this, since the state of the lamp at t = 1 cannot be logically determined by the preceding states.[6]

Modern literature

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Most of the modern literature comes from the descendants of Benacerraf, those who tacitly accept the possibility of supertasks. Philosophers who reject their possibility tend not to reject them on grounds such as Thomson's but because they have qualms with the notion of infinity itself. Of course there are exceptions. For example, McLaughlin claims that Thomson's lamp is inconsistent if it is analyzed with internal set theory, a variant of real analysis.

Philosophy of mathematics

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If supertasks are possible, then the truth or falsehood of unknown propositions of number theory, such as Goldbach's conjecture, or even undecidable propositions could be determined in a finite amount of time by a brute-force search of the set of all natural numbers. This would, however, be in contradiction with the Church–Turing thesis. Some have argued this poses a problem for intuitionism, since the intuitionist must distinguish between things that cannot in fact be proven (because they are too long or complicated; for example Boolos's "Curious Inference"[7]) but nonetheless are considered "provable", and those which are provable by infinite brute force in the above sense.

Physical possibility

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Some have claimed, Thomson's lamp is physically impossible since it must have parts moving at speeds faster than the speed of light (e.g., the lamp switch). Adolf Grünbaum suggests that the lamp could have a strip of wire which, when lifted, disrupts the circuit and turns off the lamp; this strip could then be lifted by a smaller distance each time the lamp is to be turned off, maintaining a constant velocity.

However, such a design would ultimately fail, as eventually the distance between the contacts would be so small as to allow electrons to jump the gap, preventing the circuit from being broken at all. Still, for either a human or any device, to perceive or act upon the state of the lamp some measurement has to be done, for example the light from the lamp would have to reach an eye or a sensor.

Any such measurement will take a fixed frame of time, no matter how small and, therefore, at some point measurement of the state will be impossible. Since the state at t=1 cannot be determined even in principle, it is not meaningful to speak of the lamp being either on or off.

Other physically possible supertasks have been suggested. In one proposal, one person (or entity) counts upward from 1, taking an infinite amount of time, while another person observes this from a frame of reference where this occurs in a finite space of time. For the counter, this is not a supertask, but for the observer, it is. (This could theoretically occur due to time dilation, for example if the observer were falling into a black hole while observing a counter whose position is fixed relative to the singularity.)

Gustavo E. Romero in the paper 'The collapse of supertasks'[8] maintains that any attempt to carry out a supertask will result in the formation of a black hole, making supertasks physically impossible.

Super Turing machines

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Main article: Hypercomputation

The impact of supertasks on theoretical computer science has triggered some new and interesting work, for example Hamkins and Lewis – "Infinite Time Turing Machine".[9]

Prominent supertasks

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Ross–Littlewood paradox

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Main article: Ross–Littlewood paradox

Suppose there is a jar capable of containing infinitely many marbles and an infinite collection of marbles labelled 1, 2, 3, and so on. At time t = 0, marbles 1 through 10 are placed in the jar and marble 1 is taken out. At t = 0.5, marbles 11 through 20 are placed in the jar and marble 2 is taken out; at t = 0.75, marbles 21 through 30 are put in the jar and marble 3 is taken out; and in general at time t = 1 − 0.5n, marbles 10n + 1 through 10n + 10 are placed in the jar and marble n + 1 is taken out. How many marbles are in the jar at time t = 1?

One argument states that there should be infinitely many marbles in the jar, because at each step before t = 1 the number of marbles increases from the previous step and does so unboundedly. A second argument, however, shows that the jar is empty. Consider the following argument: if the jar is non-empty, then there must be a marble in the jar. Let us say that that marble is labeled with the number n. But at time t = 1 − 0.5n - 1, the nth marble has been taken out, so marble n cannot be in the jar. This is a contradiction, so the jar must be empty. The Ross–Littlewood paradox is that here we have two seemingly perfectly good arguments with completely opposite conclusions.

Benardete's paradox

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There has been considerable interest in J. A. Benardete’s “Paradox of the Gods”:[10]

A man walks a mile from a point α. But there is an infinity of gods each of whom, unknown to the others, intends to obstruct him. One of them will raise a barrier to stop his further advance if he reaches the half-mile point, a second if he reaches the quarter-mile point, a third if he goes one-eighth of a mile, and so on ad infinitum. So he cannot even get started, because however short a distance he travels he will already have been stopped by a barrier. But in that case no barrier will rise, so that there is nothing to stop him setting off. He has been forced to stay where he is by the mere unfulfilled intentions of the gods.[11]

— M. Clark, Paradoxes from A to Z

Grim Reaper paradox

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Inspired by J. A. Benardete’s paradox regarding an infinite series of assassins,[12] David Chalmers describes the paradox as follows:

There are countably many grim reapers, one for every positive integer. Grim reaper 1 is disposed to kill you with a scythe at 1pm, if and only if you are still alive then (otherwise his scythe remains immobile throughout), taking 30 minutes about it. Grim reaper 2 is disposed to kill you with a scythe at 12:30 pm, if and only if you are still alive then, taking 15 minutes about it. Grim reaper 3 is disposed to kill you with a scythe at 12:15 pm, and so on. You are still alive just before 12pm, you can only die through the motion of a grim reaper’s scythe, and once dead you stay dead. On the face of it, this situation seems conceivable — each reaper seems conceivable individually and intrinsically, and it seems reasonable to combine distinct individuals with distinct intrinsic properties into one situation. But a little reflection reveals that the situation as described is contradictory. I cannot survive to any moment past 12pm (a grim reaper would get me first), but I cannot be killed (for grim reaper n to kill me, I must have survived grim reaper n+1, which is impossible).[13]

It has gained significance in philosophy via its use in arguing for a finite past, thereby bearing relevance to the Kalam cosmological argument.[14][15][16][17]

Davies' super-machine

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Proposed by E. Brian Davies,[18] this is a machine that can, in the space of half an hour, create an exact replica of itself that is half its size and capable of twice its replication speed. This replica will in turn create an even faster version of itself with the same specifications, resulting in a supertask that finishes after an hour. If, additionally, the machines create a communication link between parent and child machine that yields successively faster bandwidth and the machines are capable of simple arithmetic, the machines can be used to perform brute-force proofs of unknown conjectures. However, Davies also points out that – due to fundamental properties of the real universe such as quantum mechanics, thermal noise and information theory – his machine cannot actually be built.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Supertask

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A supertask is a hypothetical process consisting of a countably infinite of discrete actions or operations that are completed sequentially within a finite duration of time. The concept challenges intuitions about , time, and completion, as the infinite steps must accelerate such that their durations sum to a finite total, often following a like the with ratio 1/2. The idea of supertasks traces back to ancient paradoxes, particularly those posed by Zeno of Elea in the fifth century BCE, such as the dichotomy paradox where a traveler must traverse infinitely many subintervals to cover a finite distance. The modern term "supertask" was coined by philosopher James F. Thomson in his 1954 paper "Tasks and Super-Tasks," where he argued that such processes lead to inherent impossibilities due to ambiguities in their final states. Since then, supertasks have been analyzed in philosophy, mathematics, and physics, with key contributions from figures like Max Black (1950–51) and collections such as Wesley Salmon's Zeno's Paradoxes (1970), which formalized their discussion. Prominent examples illustrate the paradoxes arising from supertasks. In , a lamp is toggled on and off at intervals halving toward a limit time (e.g., 2 minutes total), raising the question of whether it is on or off at the endpoint, as no last toggle occurs. The Ross-Littlewood paradox involves adding and removing balls from a vase in infinite steps (10 added, 1 removed per step, accelerating), resulting in an empty vase at the end despite net additions. Other cases include Hilbert's infinite hotel, where guests shift rooms infinitely to accommodate new arrivals, and relativistic spacetimes like Malament-Hogarth structures, which permit infinite computation in finite for observers. Philosophical debates center on whether supertasks are logically coherent, physically realizable, or merely conceptual tools for exploring . Critics like Thomson contend they produce contradictions, such as indeterminate final states, while proponents argue stem from additional assumptions like continuity rather than itself. In physics, quantum and relativistic variants suggest supertasks could enable , though empirical constraints like the limit classical implementations. These discussions continue to influence fields from metaphysics to .

Definition and Fundamentals

Core Definition

A supertask is defined as a procedure consisting of a countable of distinct tasks or operations performed in succession, where the durations of these tasks form a such that the entire infinite sequence completes in a finite amount of time. This concept, introduced by philosopher James F. Thomson, emphasizes the possibility of executing infinitely many steps without extending into infinite duration, relying on progressively shortening intervals between tasks. The mathematical foundation for such completion lies in convergent infinite series, particularly . For example, consider tasks performed at times tn=∑k=1n12k=1−12nt_n = \sum_{k=1}^n \frac{1}{2^k} = 1 - \frac{1}{2^n} for n=1,2,[3,… ](/page/3Dots)n = 1, 2, [3, \dots](/page/3_Dots), approaching the limit t=1t = 1 as n→∞n \to \infty; the total time elapsed is the sum of the intervals 12+14+18+⋯=∑n=1∞(12)n=1\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = \sum_{n=1}^\infty \left( \frac{1}{2} \right)^n = 1. Here, the series converges because ∣r∣=12<1|r| = \frac{1}{2} < 1, with the general formula ∑n=1∞rn=r1−r\sum_{n=1}^\infty r^n = \frac{r}{1-r} for ∣r∣<1|r| < 1. This acceleration of task timing distinguishes supertasks from unbounded infinite processes, which do not terminate in finite time. A key conceptual challenge in supertasks is the "task completion problem," which questions the determinate state of the system at the exact limit point (e.g., precisely at t=1t = 1) following the infinite sequence, as there is no final task to establish the outcome. This issue highlights potential indeterminacies arising from the transition from finite partial sequences to the completed supertask.

Mathematical Formulation

A supertask is formally defined as a sequence of states given by a function f:N→Af: \mathbb{N} \to A, where N\mathbb{N} denotes the natural numbers and AA is a state space (such as a set of possible configurations of a system), paired with a strictly increasing sequence of timestamps tnt_n satisfying t1<t2<⋯<Tt_1 < t_2 < \dots < T, lim⁡n→∞tn=T\lim_{n \to \infty} t_n = T for some finite time TT, and ∑n=1∞(tn+1−tn)<∞\sum_{n=1}^\infty (t_{n+1} - t_n) < \infty. This ensures that infinitely many operations occur sequentially before the finite deadline TT, with the total elapsed time remaining bounded. The state at completion is then ideally lim⁡n→∞f(n)\lim_{n \to \infty} f(n), provided the limit exists in AA. The role of limits and convergence is central to this formulation, as the supremum of the completion times must be finite to model a "completed" infinite process. The sequence tnt_n converges to TT in the topological sense: for every ϵ>0\epsilon > 0, there exists N∈NN \in \mathbb{N} such that for all n>Nn > N, ∣tn−T∣<ϵ|t_n - T| < \epsilon. This ϵ\epsilon-δ\delta definition (or equivalent for general topologies) guarantees that after finitely many steps, all subsequent operations occur within any arbitrarily small interval before TT, while the convergence of the series ∑(tn+1−tn)\sum (t_{n+1} - t_n) ensures the aggregate duration is finite, often via comparison to a geometric series like ∑(1/2)n=1\sum (1/2)^n = 1. Without such convergence, the process would extend indefinitely, precluding a supertask. Non-standard analysis extends this framework by incorporating hyperreal numbers ∗R*\mathbb{R}, which include infinitesimals and infinite integers, to model supertask dynamics more intuitively. Here, time steps can be taken as infinitesimal δ≈0\delta \approx 0 (e.g., δ=1/H\delta = 1/H for infinite hypernatural HH), allowing an infinite number of steps ∑k=1Hδ=1\sum_{k=1}^H \delta = 1 (finite in the standard part) while preserving sequentiality. This approach resolves some ambiguities in standard real analysis by treating the "instant" at TT as a hyperreal point encompassing infinitesimal intervals post-convergence. The resolvability of a supertask—specifically, the existence of a well-defined limit state lim⁡n→∞f(n)\lim_{n \to \infty} f(n)—depends on properties of the state space AA. For instance, if AA is a compact metric space, every infinite sequence in AA admits a convergent subsequence by the Bolzano-Weierstrass theorem, and under additional conditions like uniform continuity of the state transitions, the full sequence converges to a unique limit in AA. However, without compactness or completeness, the limit may fail to exist or be indeterminate, as the pre-limit states do not necessarily determine a unique post-limit configuration.

Historical Origins

Zeno's Paradoxes

Zeno of Elea (c. 490–430 BCE) was a pre-Socratic philosopher associated with the Eleatic school, renowned for devising paradoxes that rigorously challenged the infinite divisibility of space and time, as well as the reality of motion and plurality. These arguments, preserved primarily through Aristotle's accounts, aimed to defend the Eleatic doctrine that the universe is a static, unchanging whole, rejecting sensory perceptions of change as illusory. The Eleatic school, originating with Xenophanes around 570 BCE and systematized by Parmenides (c. 515–450 BCE), emphasized a monistic ontology where true being is eternal and indivisible, denying the possibility of motion, becoming, or multiple entities as logical contradictions. Zeno, as a disciple of Parmenides, employed dialectical reasoning to expose inconsistencies in pluralistic views, using reductio ad absurdum to argue that accepting motion leads to paradoxes, thereby supporting the school's immaterialist and anti-empiricist stance. The Dichotomy paradox asserts that motion is impossible because, to travel a finite distance—say, from point A to B—an object must first traverse half the distance, then half of the remaining half, and so forth, resulting in an infinite sequence of subtasks that cannot be completed in finite time, despite each subinterval requiring a positive duration. This structure highlights the problem of infinite divisibility: the total time for these infinite halves sums to the full journey time, yet the infinite steps appear unachievable. Similarly, the Achilles and the tortoise paradox illustrates the same issue through a pursuit scenario: a swift runner like Achilles, pursuing a slower tortoise with a head start, must first reach the tortoise's initial position, but by then the tortoise has advanced a bit further, necessitating another catch-up, ad infinitum, such that Achilles never overtakes despite the finite overall time required. Here, the infinite intervals decrease geometrically, underscoring a supertask-like infinite regress within bounded time, challenging the coherence of continuous motion in divisible space. The Arrow paradox extends this critique to instantaneous states: at any given moment, an arrow in flight occupies a determinate space equal to its length and is thus motionless relative to that space; if time consists of such indivisible instants, the aggregation of these static moments precludes any motion whatsoever, implying that all apparent change is illusory. This argument prefigures concerns over infinite state transitions in supertasks, as it questions how discrete rests can compose continuous action. In contemporary terms, the Dichotomy and Achilles paradoxes find resolution via convergent infinite series, where the sum of diminishing terms equals a finite value, though philosophical debates persist on the physical realizability of such infinities.

Early Modern Interpretations

In the early modern period, Galileo Galilei provided one of the first systematic engagements with infinite quantities in the context of continuous motion and divisibility, ideas resonant with Zeno's ancient paradoxes. In his Dialogues Concerning Two New Sciences (1638), Galileo argued for the infinite divisibility of material continua, such as lines and solids, while acknowledging the counterintuitive nature of infinite collections. He illustrated this through a paradox involving matching infinities: the set of natural numbers can be put into one-to-one correspondence with the set of their squares, despite the latter being a proper subset of the former, suggesting that traditional notions of size fail for infinities. This handling of infinite enumerations prefigured concerns in supertasks by demonstrating how infinite processes could yield paradoxical equalities without physical completion. Gottfried Wilhelm Leibniz extended these ideas in the late 17th century by developing infinitesimal calculus as a tool for modeling infinite processes within finite durations, offering a metaphysical and mathematical framework for continuous change. In works like (1714), Leibniz posited monads—simple, indivisible substances—as the fundamental units of reality, each perceiving the universe through an infinite series of states unfolding continuously over time, governed by the Principle of Continuity. His calculus treated infinitesimals as syncategorematic fictions: useful approximations for derivatives and integrals that resolve apparent discontinuities in motion, such as those implied by infinite subdivisions of space or time. This approach implicitly addressed supertask-like infinities by allowing infinite approximations to converge in finite steps, bridging philosophical puzzles of indivisibility with practical computation. George Berkeley mounted a significant critique in the 18th century, rejecting Leibnizian infinitesimals as philosophically incoherent and fueling skepticism toward infinite processes. In The Analyst (1734), Berkeley derided these quantities as "ghosts of departed quantities," arguing they oscillate inconsistently between zero and non-zero without clear ontological status, thus undermining the foundations of calculus used to handle infinite divisions. His attack highlighted the risks of assuming infinite tasks could be meaningfully "performed" in continua, influencing later caution in interpreting supertasks as actual sequences rather than ideal limits. The 19th century saw a pivotal shift toward rigorization, with Augustin-Louis Cauchy providing a rigorous treatment of limits in his Cours d'analyse (1821), which demonstrated the convergence of infinite sums, such as geometric series representing successive halvings in the dichotomy paradox, to finite values without invoking actual infinities. Karl Weierstrass further refined this framework in the 1850s–1860s by introducing epsilon-delta definitions for limits, continuity, and derivatives, ensuring the real number line has no gaps and allowing infinite processes to be analyzed as completed wholes in finite time. This development implicitly dissolved supertask paradoxes by showing that infinite divisions need not imply physical impossibility, paving the way for 20th-century revivals in philosophy and logic.

Formative 20th-Century Contributions

Max Black's Analysis

In his 1951 paper "Achilles and the Tortoise," Max Black provided an early modern analysis of supertasks by examining Zeno's paradox of Achilles and the tortoise. Black accepted that the infinite series of distances sums to a finite total but argued that completing the supertask remains impossible because the sequence has no final step. Without a last action to conclude the process, the task cannot be coherently finished, leading to a logical incoherence in actual infinities. This emphasis on the absence of a terminating action in infinite sequences influenced subsequent discussions, including Thomson's later formulation of supertasks.

Thomson's Lamp Paradox

James F. Thomson introduced the concept of a supertask in his 1954 paper to critique the logical possibility of completing an infinite number of tasks within a finite duration, arguing that such processes involving actual infinities in time lead to absurdities. Motivated by Zeno's paradoxes and Bertrand Russell's suggestions about accelerating task performance, Thomson sought to demonstrate that supertasks cannot be coherently described or realized. The paradox centers on a lamp that starts in the off position at time t=0. It is toggled on at t=0.5 minutes, off at t=0.75 minutes, on at t=0.875 minutes, and so forth, with each subsequent toggle occurring after half the previous interval, completing infinitely many switches by t=1 minute. The times of these toggles form a convergent series summing to 1 minute, mirroring the mathematical structure of supertasks where infinite steps accumulate in finite time. At t=1 minute, the lamp's state is indeterminate: it cannot be on, as every instance of it being on is followed by a turn-off, yet it cannot be off, as every turn-off is followed by a turn-on, with no final toggle to settle the matter. Thomson argued that this ambiguity reveals a fundamental problem with supertasks: they require a well-defined limit state after the infinite sequence, but the absence of a "last" task makes such specification impossible, rendering the entire process conceptually incoherent. He contended that supertasks thus fail to provide a viable model for actual infinities, as the lamp's undefined state at the completion time underscores the impracticality of infinite temporal divisions in real processes. This formulation revived philosophical interest in infinite processes, sparking ongoing debates about whether supertasks are logically possible constructs or merely descriptive fictions that break down under scrutiny. A variant considers symmetric toggling without presupposing an initial state, which amplifies the indeterminacy by avoiding any baseline for the sequence.

Benacerraf's Analysis

In his 1962 paper, Paul Benacerraf provided a formal analysis of supertasks by rephrasing James F. Thomson's lamp paradox in mathematical terms, modeling the infinite sequence of actions as a function defined over the ordinal ω\omega, the smallest infinite ordinal in set theory. This setup represents the supertask as a countable sequence of discrete steps approaching a limit time t1t_1, where each step corresponds to toggling the lamp's state at intervals halving in duration (e.g., 1 minute, then 1/2 minute, and so on), forming a series of order type ω\omega. Benacerraf emphasized that such sequences, unlike finite tasks, do not inherently specify a unique completion state at the limit, as the infinite history alone fails to determine the outcome without additional assumptions. A central issue in Benacerraf's critique is the underdetermination of the final state, illustrated by the possibility of multiple compatible descriptions of the same supertask yielding contradictory results. For instance, one can describe the lamp as "on" at t1t_1 by considering only the even-numbered steps (where it ends up on after each pair of toggles), or "off" at t1t_1 by focusing on the odd-numbered steps (ending off after each initial toggle). This ambiguity arises because the supertask's sequence, while well-defined over ω\omega, lacks a canonical extension to the limit point; no logical rule mandates that the state at t1t_1 must follow directly from the infinite prior actions, as Thomson assumed. Benacerraf noted that Thomson's argument for impossibility relies on an unproven analytic premise—that the lamp's state at t1t_1 is analytically determined by its states before t1t_1—which classical mathematics does not guarantee for infinite processes. Benacerraf's analysis highlights broader implications for logic and classical mathematics, challenging the coherence of supertasks by showing how they expose ambiguities in handling infinite sequences without leading to outright contradiction. In set theory, this connects to the distinction between sequences indexed by ordinals like ω\omega (discrete and countable) and continuous structures, underscoring that supertasks as proper classes or transfinite sequences do not resolve determinacy issues merely through formalization. Ultimately, Benacerraf argued that the paradoxes stem not from impossibility but from incomplete specifications of what constitutes "completion" in infinite tasks.

Prominent Paradoxes and Examples

Ross-Littlewood Vase Paradox

The Ross-Littlewood vase paradox originated as an informal puzzle posed by mathematician J. E. Littlewood in his 1953 collection A Mathematician's Miscellany, where it was presented as an example of counterintuitive infinity in mathematics. It was later formalized and analyzed in greater detail by Sheldon M. Ross in the 1988 edition of his textbook A First Course in Probability, which explored its implications for infinite processes. The paradox serves as a classic illustration of a supertask, involving infinitely many operations completed in a finite duration. The setup begins with an empty vase and an infinite supply of distinctively numbered balls, labeled with natural numbers starting from 1. The process unfolds over a finite time interval, such as the hour leading to noon, with each step nn occurring at time tn=1−12n−1t_n = 1 - \frac{1}{2^{n-1}} minutes before noon (ensuring the steps converge to noon). At step nn, ten new balls numbered 10(n−1)+110(n-1) + 1 through 10n10n are added to the vase, increasing its contents by ten. Immediately after, the ball with the lowest number currently in the vase—specifically, ball nn—is removed. This net addition of nine balls repeats infinitely many times as nn approaches infinity, with the entire supertask completing exactly at noon. The paradox arises from conflicting intuitions about the vase's state at noon. On one hand, since nine balls are added net at each of the infinitely many steps, one might expect the vase to contain infinitely many balls at the limit, as the cardinality after nn steps is exactly 9n9n, which diverges to infinity as n→∞n \to \infty. On the other hand, a closer examination reveals that every individual ball kk is added at some finite step (specifically, during step ⌈k/10⌉\lceil k/10 \rceil) and removed at a later finite step kk, leaving no ball present at noon. This diagonal-style argument shows that the vase must be empty, as for any purported ball mm in the final state, there exists a finite step mm after which it is absent. Variants of the paradox highlight how the outcome depends on the removal rule, demonstrating the sensitivity of supertasks to procedural details. In the standard version above, removing the lowest-numbered ball leads to an empty vase. However, if at each step nn the removal instead targets one ball from the most recently added group (e.g., always the first of the ten just introduced), then infinitely many balls remain at noon—specifically, the second through tenth balls from each batch, forming a countably infinite set. Another variant alters the removal rate: at step nn, ten balls are added, but nn balls (the lowest-numbered ones) are removed, yielding a net change of 10−n10 - n. While early steps increase the count, later steps with n>10n > 10 would imply negative net additions, rendering the process ill-defined beyond certain points; analyses of such "slower" removals often result in finite or infinite residues depending on labeling and timing, underscoring the non-uniqueness of limits in supertasks. Logically, the presents no true contradiction, as the apparent conflict stems from conflating the limit of the sequence of states with the state of the limit. The number of balls grows without bound during the process (lim⁡n→∞∣Vn∣=∞\lim_{n\to\infty} |V_n| = \infty, where VnV_n is the vase after nn steps), but the final configuration at noon is the set of balls that are never removed, which is empty in the standard case. This distinction aligns with set-theoretic principles, where infinite unions or intersections do not commute with measures, resolving the via precise mathematical limits rather than physical .

Benardete's Dichotomy Paradox

Benardete's dichotomy paradox, introduced by philosopher José A. Benardete, presents a supertask scenario involving a man attempting to walk along a road of finite length, say one mile from point A to point B, where an infinite sequence of divine interventions determines the presence of barriers at points approaching B. Specifically, for each n, there is a point xn=1−12nx_n = 1 - \frac{1}{2^n} along the road (e.g., x1=1/2x_1 = 1/2, x2=3/4x_2 = 3/4, x3=7/8x_3 = 7/8, and so on, converging to 1). At each such point, a pair of gods is stationed: one god tasked with erecting an impenetrable barrier at xnx_n if the man ever reaches xnx_n, ensuring his death by collision, and the other god tasked with removing any such barrier if the man never reaches xnx_n. This setup creates a chain of conditional decisions, where the action at each point depends on whether the man arrives there, which in turn depends on the absence of prior barriers. The paradoxical outcome arises from evaluating the global state after all infinite interventions. Since the points xnx_n form a convergent sequence with no final point, the man would reach every xnx_n only if no barrier blocks him before it, but the conditionals ensure that if he reaches any xnx_n, a barrier appears there to stop him. Consequently, no barrier is ever erected, as the assumption that he reaches any particular xnx_n leads to a contradiction (he would be stopped there), so all gods remove their barriers, allowing the man to pass every point and reach B unscathed. However, this global survival contradicts an infinite disjunction derived from the local conditionals: the man must die, because either a barrier appears at x1x_1 (stopping him at 1/21/2), or if not, at x2x_2 (stopping him at 3/43/4), or if not, at x3x_3 (at 7/87/8), and so on for every n—implying he is stopped at some finite stage, yet no such stage exists. This tension highlights a survivor scenario where the man lives through the supertask, but the logic of the disjunction demands his demise. The paradox underscores themes of in , where each conditional action presupposes the outcomes of infinitely many prior ones, and in the metaphysics of conditionals, questioning how counterfactual dependencies behave under supertasks. Benardete frames this as a modern variant of Zeno's dichotomy, but with intentional agents introducing causal loops that render the supertask incoherent. Proposed resolutions often argue that such supertasks are impossible, as the infinite chain of conditionals creates irresolvable causal dependencies, preventing any consistent execution.

Grim Reaper Paradox

The Grim Reaper paradox, originally formulated by philosopher José A. Benardete in 1964 as the "assassin paradox," presents a supertask scenario that challenges the coherence of infinite causal sequences converging in finite time. In this setup, an immortal man named Fred stands at noon (t=0t=0), with infinitely many grim reapers poised to act at successively earlier times approaching t=0t=0 from the positive side. The nth reaper is instructed to awaken exactly at time tn=1nt_n = \frac{1}{n} (for positive integers n) and, upon finding Fred alive, to kill him instantly; if Fred is already dead, the reaper does nothing and remains inactive. Popularized in late 20th- and early 21st-century supertask literature, including works by and Robert Koons, the paradox highlights tensions in actual infinities and backward infinite causal chains. By t=0t=0, every has had its opportunity to act, yet Fred appears to survive because no single kills him: for any particular n, there are infinitely many subsequent reapers (with higher n) that would have acted earlier if Fred were still alive at their times, but the setup ensures an where each misses its chance due to temporal gaps. However, Fred cannot possibly be alive at t=0t=0, as the with the latest activation time (smallest n) would find him alive and kill him, presupposing survival through all prior infinite activations. This leads to the core of : if Fred is dead at t=0t=0, then some must have killed him, but identifying which one requires an impossible infinite descent in the causal chain, implying he died infinitely often without a first cause; if alive, the entire infinite sequence fails to achieve its deterministic outcome. The scenario thus generates an acausal result—death without a specific killer—or violates temporal order by requiring causation without temporal precedence. Variants of the paradox extend these issues. In synchronic versions, infinitely many reapers are spatially arranged (e.g., at distances dn=2−nd_n = 2^{-n} from Fred) and activate simultaneously at t=0t=0, with each killing only if no prior (closer) reaper has acted, yielding the same underdetermination: Fred dies without a determinate cause. Probabilistic variants introduce chance, such as each reaper killing with probability p<1p < 1 if alive, where the infinite product of survival probabilities approaches zero, yet the deterministic structure still forces survival at t=0t=0, amplifying the apparent violation of expected outcomes in supertasks. These formulations, akin in conditional structure to Benardete's spatial dichotomy paradox of rising barriers, underscore how supertasks can produce outcomes defying intuitive causality.

Davies' Supertask Device

In 2001, physicist proposed a physical realization of a supertask device in his book How to Build a Time Machine, envisioning a "supermachine" that could execute an infinite sequence of computations within a finite timeframe by exploiting and relativistic effects. This concept draws inspiration from paradox, adapting its infinite toggling into a tangible mechanism. The device's setup centers on a quantum system that repeatedly flips between states, with the operation rate accelerating dramatically as energies approach the Planck scale (approximately 10−3510^{-35} meters and 10−4310^{-43} seconds). As the process intensifies, —arising from the high energies or proximity to massive objects—progressively slows the perceived passage of time for the system relative to an external observer, effectively compressing the infinite flips into a bounded interval. emphasized that this compression allows the supertask to conclude in finite laboratory time while the infinity remains "hidden" behind an or within a structure, consistent with general relativity's allowance for such geometries. Davies argued that the supermachine's feasibility hinges on these relativistic features, positing it as a bridge between abstract philosophical paradoxes and practical scientific engineering. However, critics have highlighted thermodynamic issues, noting that the accelerating energy requirements could violate the second law by implying perpetual motion-like efficiency in state transitions. Additionally, quantum measurement limits—such as the uncertainty principle—may prevent precise control over the state flips at Planck scales, rendering the infinite sequence practically unattainable. This proposal holds significance as the first explicit outline of a physically realizable supertask following Thomson's paradox, shifting discussions from pure logic to testable physics despite the challenges.

Contemporary Perspectives

Philosophical Debates

Philosophical debates surrounding supertasks since the have primarily focused on their metaphysical foundations, questioning whether such processes are coherent given concepts of , temporality, and agency. Critics argue that supertasks presuppose an actual infinity— a completed infinite —rather than a mere potential infinity, which unfolds indefinitely without completion. This distinction revives Aristotelian concerns, as actual infinities are seen as leading to absurdities, such as infinite magnitudes or of division, whereas potential infinities align with finite, processual reality. A related contention involves the nature of time and change, where supertasks appear to demand a static B-series ordering of events (earlier-than and later-than relations fixed eternally) while conflicting with the dynamic A-series of tensed experience (past, present, future flow). This tension echoes McTaggart's paradox, suggesting that supertasks undermine the intuitive passage of time by compressing infinite changes into a finite interval, rendering the "aftermath" of the supertask indeterminate or illusory. John Earman has defended the logical possibility of supertasks within classical frameworks, asserting they pose no formal contradiction but strain human intuition about temporal continuity and completion. Debates also extend to and , particularly in variants like Benardete's infinite decision scenarios, where an agent or series of agents must make infinitely many choices in finite time, potentially eroding agency by forcing predetermined outcomes or undecidable states. Such supertasks challenge compatibilist views, as the of decisions implies a loss of autonomous control, aligning with deterministic chains that preclude genuine choice. Recent contributions, such as Jon Pérez Laraudogoitia's analysis of a reversible supertask involving infinite particle collisions, further intensify these issues by demonstrating how supertasks could ostensibly create entities (like particles) ex nihilo through symmetric time-reversal, questioning conservation principles in metaphysical terms without resolving underlying logical tensions.

Physical and Scientific Feasibility

In , supertasks appear feasible in certain pathological spacetimes known as Malament-Hogarth spacetimes, where near a rotating black hole's allows an observer to experience an infinite for performing the task while a distant observer perceives completion in finite . However, this setup demands infinite for the agent's to avoid falling into the horizon, and the resulting unbounded gravitational forces would physically destroy any realizable device, rendering the supertask unattainable. Proposals like ' supertask machine, which envisions a self-replicating device using accelerating classical processes, face similar barriers due to the infinite resources required. Quantum mechanics imposes fundamental limits on supertasks through principles that preclude the infinite precision and state manipulation needed for their execution. The Heisenberg , ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}, restricts simultaneous knowledge of position and momentum, making it impossible to localize and accelerate components with the exactitude demanded by successively faster operations in a supertask. In quantum supertask variants involving repeated state preparations or measurements, the further blocks perfect replication of arbitrary unknown quantum states, as unitary evolution cannot produce identical copies without disturbing the original. These constraints attempts at quantum supertasks into singularities or indeterminate outcomes long before is reached. Thermodynamic considerations reveal additional barriers, as infinite operations in finite time would generate unbounded , violating the second law of , which mandates that in an cannot decrease and increases at a finite rate. Each step, even idealized, dissipates as heat, leading to an infinite total entropy rise incompatible with closed-system constraints in any finite interval. No direct experimental realizations of supertasks exist, but the serves as a partial analog, where frequent projective measurements inhibit evolution, effectively "freezing" the system as if intervening infinitely often. Observed in experiments with trapped ions, this effect demonstrates slowed decay rates under repeated observations, mirroring the disruptive interventions of a supertask but without achieving infinite steps. In 21st-century cosmology, Frank Tipler's theory posits supertasks at the 's end, where a collapsing closed enables infinite computational operations approaching the final singularity, potentially resurrecting all via reversible physics. However, this remains speculative, reliant on unverified assumptions about cosmic closure and , and conflicts with observational evidence for accelerating expansion.

Computational Implications

In , supertasks have been explored as mechanisms for , enabling models that surpass the limitations of by completing infinitely many steps within finite time. These concepts emerged prominently in the late , building on earlier ideas like Turing's machines from 1939, which hypothetically access external information to solve undecidable problems, but supertask variants emphasize temporal acceleration to achieve similar ends without oracles. Accelerating , proposed by in 1998, exemplify this approach: they follow the standard architecture but execute steps at rates that converge to a supertask, such as performing computations in Zeno-like time intervals summing to a finite duration, potentially allowing resolution of uncomputable functions like determining whether a halts on a given input. One influential model integrates supertasks with through Malament-Hogarth spacetimes, developed in the by David Malament and Mark Hogarth. In these spacetimes, an observer along a timelike curve of infinite can perform an infinite sequence of computational steps, while a distant observer experiences only finite until a signal from the completion point arrives. This setup, which does not require closed timelike curves, permits by simulating infinite tapes or iterations in observer time, theoretically deciding problems like the through exhaustive simulation. Proponents argue that such supertasks could solve the , an undecidable question in Turing computability, by encoding the machine and input into an accelerating process that runs indefinitely along the infinite curve and outputs a result at the finite endpoint if halting occurs. However, critics contend that supertasks do not inherently expand computational power; for instance, accelerating machines without a well-defined limit stage fail to reliably output solutions to uncomputable functions, mirroring paradoxes like where the final state remains indeterminate. Physically, these models face unrealizability: Malament-Hogarth spacetimes violate global hyperbolicity, leading to instabilities, and no known solutions to Einstein's field equations support them without invoking speculative conditions that contradict the . More recent proposals have examined quantum supertasks within adiabatic computing frameworks, where slow evolution through a quantum might approximate infinite-step processes for solving non-recursive problems like Hilbert's tenth. For example, Tien D. Kieu's 2004 algorithm uses the quantum to navigate energy landscapes corresponding to Diophantine equations, claiming al capability. Yet, these are sharply criticized for conflating quantum parallelism with true hypercomputation, as they remain bounded by recursive definitions and fail to output noncomputable results reliably. Moreover, practical limitations arise from decoherence, where environmental interactions disrupt the adiabatic condition, preventing sustained infinite-like evolutions in finite time and confining quantum adiabatic models to polynomial-time approximations of Turing-computable functions.

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