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- §1: What is a Supertask
- Definitions
- The philosophical problem of supertasks
- Supertask: A Fuzzy Concept

- §2: On the Conceptual Possibility of Supertasks
- Zeno’s Dichotomy Paradox
- The Inverse Form of the Dichotomy Argument
- On Thomson’s Impossibility Arguments
- On Black’s Impossibility Argument
- Benacerraf’s Critique and Zeno’s Dichotomy Arguments
- Conclusion

- §3: On the Physical Possibility of Supertasks
- Kinematical Impossibility
- The Principle of Continuity and the Solution to the Philosophical Problem of Supertasks
- The Postulate of Permanence

- §4: The Physics of Supertasks
- A New Form of Indeterminism: Spontaneous Self-Excitation
- Bifurcated Supertasks
- Bifurcated Supertasks and the Solution to the Philosophical Problem of Supertasks

- §5: What Supertasks Entail for the Philosophy of Mathematics
- A Critique of Intuitionism
- The Importance of the Malament-Hogarth Spacetime

- Bibliography
- Other Internet Resources
- Related Entries

There is a particular type of supertask called *hypertasks*. A
hypertask is a non-numerable infinite sequence of actions or
operations carried out in a finite interval of time. Therefore, a
supertask which is not a hypertask will be a numerable infinite
sequence of actions or operations carried out in a finite interval of
time. Finally, a task can be defined as a finite sequence of actions
or operations carried out in a finite interval of time.

In the case of a task T = (a_{1}, a_{2},
a_{3}, . . . , a_{n}) it is natural to say that T is
applicable in state S if:

aThe successive states of the world relevant to task T can be defined by means of the finite sequence of sets of sentences:_{1}is applicable to S,

a_{2}is applicable to a_{1}(S),

a_{3}is applicable to a_{2}(a_{1}(S)),

. . . , and,

a_{n}is applicable to a_{n-1}(a_{n-2}(. . . (a_{2}(a_{1}(S))). . . )).

S, awhose last term will therefore describe the relevant state of the world after the performance of T. Or, equivalently, the state resulting from applying T to S will be T(S) =_{1}(S), a_{2}(a_{1}(S)), a_{3}(a_{2}(a_{1}(S))), . . . , a_{n}(a_{n-1}(a_{n-2}(. . . (a_{2}(a_{1}(S))). . . ))),

aNow take the case of a supertask T = (a_{n}(a_{n-1}(a_{n-2}(. . . (a_{2}(a_{1}(S))). . . ))).

TThe successive states of the world relevant to supertask T can be described by means of the infinite sequence of sets of sentences:_{n}(S) = a_{n}(a_{n-1}(a_{n-2}(. . . (a_{2}(a_{1}(S))). . . ))).

S, TA difficulty arises, however, when we want to specify the set of sentences which describe the relevant state of the world after the performance of supertask T, because the infinite sequence above lacks a final term. Put equivalently, it is difficult to specify the relevant state of the world resulting from the application of supertask T to S because there seems to be no final state resulting from such an application. This inherent difficulty is increased by the fact that, by definition, supertask T is performed in a finite time, and so there must exist one first instant of time t* at which it can be said that the performance happened. Now notice that the world must naturally be in a certain specific state at t*, which is the state resulting from the application of T, but that, nevertheless, we have serious trouble to specify this state, as we have just seen._{1}(S), T_{2}(S), . . . , T_{n}(S), . . .

As an instance of the second sort of processes we referred to above,
those about which no consensus has been reached as to whether they
are supertasks, we can take the process which is described in one of
the forms of Zeno’s dichotomy paradox. Suppose that initially
(at t = 12 A.M., say) Achilles is at point A (*x* = 0) and
moving in a straight line, with a constant velocity *v* = 1
km/h, towards point B (*x* = 1), which is 1 km. away from A.
Assume, in addition, that Achilles does not modify his velocity at
any point. In that case, we can view Achilles’s run as the
performance of a supertask, in the following way: when half the time
until t* = 1 P.M. has gone by, Achilles will have carried out the
action a_{1} of going from point *x* = 0 to point
*x* = 1/2 (a_{1} is thus performed in the interval of
time between t =12 A.M. and t = 1/2 P.M.), when half the time from
the end of the performance of a_{1} until t* = 1 P.M. will
have elapsed, Achilles will have carried out the action a_{2}
of going from point *x* = 1/2 to point *x* = 1/2 + 1/4
(a_{2} is thus performed in the interval of time between t =
1/2 P.M. and t = 1/2 + 1/4 P.M.), when half the time from the end of
the performance of a_{2} until t* = 1 P.M. will have elapsed,
Achilles will have carried out the action a_{3} of going from
point *x* = 1/2 + 1/4 to point *x* = 1/2 + 1/4 + 1/8
(a_{3} is thus performed in the interval of time between t =
1/2 + 1/4 P.M. and t = 1/2 + 1/4 + 1/8 P.M.), and so on. When we get
to instant t* = 1 P.M., Achilles will have carried out an infinite
sequence of actions, that is, a supertask T = (a_{1},
a_{2}, a_{3}, . . . , a_{n}, . . . ),
provided we allow the state of the world relevant for the description
of T to be specified, at any arbitrary instant, by a single sentence:
the one which specifies Achilles’s position at that instant.
Several philosophers have objected to this conclusion, arguing that,
in contrast to Thomson’s lamp, Achilles’s run does not
involve an infinity of actions (acts) but of pseudo-acts. In their
view, the analysis presented above for Achilles’s run is nothing
but the breakdown of one process into a numerable infinity of
subprocesses, which does not make it into a supertask. In Allis and
Koetsier’s words, such philosophers believe that a set of
position sentences is not always to be admitted as a description of
the state of the world relevant to a certain action. In their
opinion, a relevant description of a state of the world should
normally include a different type of sentences (as is the case with
Thomson’s lamp) or, in any case, more than simply position
sentences.

In stark contrast to Zeno, the dichotomy paradox is standardly solved
by saying that the successive distances covered by Achilles as he
progressively reaches the mid points of the spans he has left to go
through --- 1/2, 1/4, 1/8, 1/16, . . . --- form an infinite
series 1/2 + 1/4 + 1/8 + 1/16 + . . . whose sum is 1.
Consequently, Achilles will indeed reach point B (*x* = 1) at t*
= 1 P.M. (which is to be expected if he travels with velocity *v* =
1 km/h, as has been assumed). Then there is no problem whatsoever in
splitting up his run into smaller sub-runs and, so, no inherent
problem about the notion of supertask. An objection can be made,
however, to this standard solution to the paradox: it tells us where
Achilles is at each instant but it does not explain where Zeno’s
argument breaks down. Importantly, there is another objection to the
standard solution, which hinges on the fact that, when it is claimed
that the infinite series 1/2 + 1/4 + 1/8 + 1/16 + . . .
adds up to 1, this is substantiated by the assertion that the sequence
of partial sums 1/2, 1/2 + 1/4, 1/2 + 1/4 +
1/8, . . . has limit 1, that is, that the difference between the
successive terms of the sequence and number 1 becomes progressively
smaller than any positive integer, no matter how small. But it might
be countered that this is just a patch up: the infinite series
1/2 + 1/4 + 1/8 + . . . seems to involve infinite sums and
thus the performance of a supertask, and the proponent of the standard
solution is in fact presupposing that supertasks are feasible just in
order to justifiy that they are. To this the latter might reply that
the assertion that the sum of the series is 1 presupposes no infinite
sum, since, by definition, the sum of a series is the limit to which
its partial (and so finite) sums approach. His opponent can now
express his disagreement with the response that the one who supports
the standard solution is deducing a matter of fact (that Achilles is
at *x* = 1 at t* = 1 P.M.) from a definition pertaining to the
arithmetic of infinite series, and that it is blatantly unacceptable
to deduce empirical propositions from mere definitions.

Thomson (1954-55) put forward one more argument against the logical
possibility of his lamp. Let us assign to the lamp the value 0 when
it is dim and the value 1 when it is lit. Then lighting the lamp
means adding one unity (going from 0 to 1) and dimming it means
subtracting one unity (going from 1 to 0). It thus seems that the
final state of the lamp at t* = 1 P.M., after an infinite, and
alternating, sequence of lightings (additions of one unity) and
dimmings (subtractions of one unity), should be described by the
infinite series 1-1+1-1+1. . . If we accept the conventional
mathematical definition of the sum of a series, this series has no
sum, because the partial sums 1, 1-1, 1-1+1, 1-1+1-1, . . . ,
etc. take on the values 1 and 0 alternatively, without ever
approaching a definite limit that could be taken to be the proper sum
of the series. But in that case it seems that the final state of the
lamp can neither be dim (0) nor lit (1), which contradicts our
assumption that the lamp is at all times either dim or lit.
Benacerraf’s (1962) reply was that even though the first, second, third,
. . . , n-th partial sum of the series 1-1+1-1+1. . . does yield the
state of the lamp after one, two, three, . . . , n actions
a_{i} (of lighting or dimming), it does not follow from this
that the final state of the lamp after the infinite sequence of
actions a_{i} must of necessity be given by the sum of the
series, that is, by the limit to which its partial sums progressively
approach. The reason is that a property shared by the partial sums of
a series does not have to be shared by the limit to which those
partial sums tend. For instance, the partial sums of the series 0.3 +
0.03 + 0.003 + 0.0003 + . . . are 0.3, 0.3 + 0.03 = 0.33, 0.3 + 0.03 +
0.003 = 0.333,. . . , all of them, clearly, numbers less than 1/3;
however, the limit to which those partial sums tend (that is, the sum
of the original series) is 0.3333... , which is precisely the number
1/3.

(I) the state of a system at an instant t* is not a logical consequence of which states he has been in before t* (where by ‘state’ I mean ‘relevant state of the world’, see section 1.1)and occasionally on the idea that

(II) the properties shared by the partial sums of a series do not have to be shared by the limit to which those partial sums tend.Since the partial sums of a series make up a succession (of partial sums), (II) may be rewritten as follows:

(III) the properties shared by the terms of a succession do not have to be shared by the limit to which that succession tends.If we keep (I), (II) and (III) well in mind, it is easy not to yield to the perplexing implications of certain supertasks dealt with in the literature. And if we do not yield to the perplexing results, we will also not fall into the trap of considering supertasks conceptually impossible. (III), for instance, may be used to show that it is not impossible for Achilles to perform the supertasks of the inverse and the direct dichotomy of Zeno. Take the case of the direct dichotomy: the limit of the corresponding succession of instants of time t

a) At least one of the moving bodies travels at an unboundedly increasing speed,It is clear then that the Thomson’s lamp supertask, in the version presented so far, is kinematically (and eo ipso physically) impossible, since not only does the moving switch have to travel at a speed that increases unboundedly but also -because it oscillates between two set positions which are a constant distance d apart- its position does not approach any definite limit as we get closer to instant t* = 1 P.M., at which the supertask is accomplished. Nevertheless, Grünbaum has also shown models of Thomson’s lamp which are kinematically possible. Take a look at Figure 1, in which the switch (in position ‘on’ there) is simply a segment AB of the circuit connecting generator G with lamp L. The circuit segment AB can shift any distance upwards so as to open the circuit in order for L to be dimmed. Imagine we push the switch successively upwards and downwards in the way illustrated in Figure 2, so that it always has the same velocityb) For some instant of time t*, the position of at least one of the moving bodies does not approach any defined limit as we get arbitrarily closer in time to t*.

Figure 1

The procedure is the following. Initially (t = 0) the switch is in position AB (lamp dim) a height of 0.2 above the circuit and moving downwards (at

Figure 2

There are some who believe that the very fact that there exist Thomson’s lamps yielding an intuitive result of ‘lamp lit’ when the supertask is accomplished but also other lamps whose intuitive result is ‘lamp dim’ brings up back to the contradiction which Thomson thought to have found originally. But we have nothing of that sort. What we do have is different physical models with different end-results. This does not contradict but rather corroborates the results obtained by Benacerraf: the final state is not logically determined by the previous sequence of states and operations. This logical indeterminacy can indeed become physical determinacy, at least sometimes, depending on what model of Thomson’s lamp is employed.

Figure 3

A conspicuous instance of a supertask which is kinematically
impossible is the one performed by Black’s infinity machine, whose
task it is to transport a ball from position A (*x* = 0) to
position B (*x* = 1) and from B to A an infinite number of times
in one hour. As with the switch in our first model of Thomson’s lamp,
it is obvious that the speed of the ball increases unboundedly (and so
condition a) for impossibility is met), while at the same time, as we
approach t* = 1 P.M., its position does not tend to any defined limit,
due to the fact that it must oscillate continuously between two set
positions A and B one unity distance apart from each other (and so
also condition b) for impossibility is met).

Since Benacerraf’s critique, we know that there is no logical
connection between the position of Achilles at t* = 1 P.M. and his
positions at instants previous to t* = 1 P.M. Sainsbury [1988] has
tried to bridge the gap opened by Benacerraf. He claims that this can
be achieved by drawing a distinction between abstract space of a
mathematical kind and physical space. No distinction between
mathematical and physical space has to be made, however, to attain
that goal; one need only appeal to a single principle of physical
nature, which is, moreover, simple and general, namely, that the
trajectories of material bodies are continuous lines. To put it more
graphically, what this means is that we can draw those trajectories
without lifting our pen off the paper. More precisely, that the
trajectory of a material body is a continuous line means that,
whatever the instant t, the limit to which the position occupied by
the body tends as time approaches t coincides precisely with the
position of the body at t. Moreover, the principle of continuity is
highly plausible as a physical hypothesis: the trajectories of all
physical bodies in the real world are in fact continuous. What
matters is that we realise that, aided by this principle, we can now
finally demonstrate that after the accomplishment of the dichotomy
supertask, that is, at t* = 1 P.M., Achilles will be in point B
(*x* = 1). We know, in fact, that as the time Achilles has spent
running gets closer and closer to t* = 1 P.M., his position will
approach point *x* = 1 more and more, or, equivalently, we know
that the limit to which the position occupied by Achilles tends as
time get progressively closer to t* = 1 P.M. is point B (*x* =
1). As Achilles’s trajectory must be continuous, by the definition of
continuity (applied to instant t = t* = 1 P.M.) we obtain that the
limit to which the position occupied by Achilles tends as time
approaches t* = 1 P.M. coincides with Achilles’s position at t* = 1
P.M. Since we also know that this limit is point B (*x* = 1), it
finally follows that Achilles’s position at t* = 1 P.M. is point B
(*x* = 1). Now is when we can spot the flaw in the standard
argument against Zeno mentioned in section 2.1, which was grounded on
the observation that the sequence of distances covered by Achilles
(1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, . . . ) has
1 as its limit. This alone does not suffice to conclude that Achilles
will reach point *x* = 1, unless it is assumed that if the
distances run by Achilles have 1 as their limit, then Achilles will as
a matter of fact reach *x* = 1, but assuming this entails using
the principle of continuity. This principle affords us a rigorous
demonstration of what, in any event, was already plausible and
intuitively ‘natural’: that after having performed the infinite
sequence of actions (a_{1}, a_{2}, a_{3},
. . . , a_{n}, . . . ) Achilles will have reached point B
(*x* = 1). In addition, now it is easy to show how, with a
switch like the one in Figure 2, Thomson’s lamp in Figure 1 will reach
t* = 2/9 with its switch in position AB and will therefore be lit. We
have in fact already pointed out (3.1) that in this case, as we get
closer to the limit time t* = 2/9, the switch indefinitely approaches
a well-defined limit position -position AB. Due to the fact that the
principle of continuity applies to the switch, because it is a
physical body, this well-defined limit position must coincide
precisely with the position of the switch at t* = 2/9. Therefore, at
t* = 2/9 the latter will be in positon AB and, consequently, the lamp
will be lit. By the same token, it can also be shown that the lamp in
Figure 3 will be dim at time t* = 2/9.

the well-known dynamic theorem by which if two identical particles undergo an elastic collision then they will exchange their velocities after colliding. If our particles P

Figure 4

The supertask literature has needed to exploit space-times with sufficiently complicated structure that global reference systems cannot be defined in them. In these and other cases, the time of a process can be represented by its ‘proper time’. If we represent a process by its world-line in space-time, the proper time of the process is the time read by a good clock that moves with the process along its world-line. A familiar example of its use is the problem of the twins in special relativity. One twin stays home on earth and grows old. Forty years of proper time, for example, elapses along his world-line. The travelling twin accelerates off into space and returns to find his sibling forty years older. But much less time -- say only a year of proper time -- will have elapsed along the travelling twin world-line if he has accelerated to sufficiently great speeds.

If we take this into account it is easily seen that the definition of supertask that we have been using is ambiguous. In section 1 above we defined a supertask as an infinite sequence of actions or operations carried out in a finite interval of time. But we have not specified in whose proper time we measure the finite interval of time. Do we take the proper time of the process under consideration? Or do we take the proper time of some observer who watches the process? It turns out that relativity theory allows the former to be infinite while the latter is finite. This fact opens new possibilities for supertasks. Relativity theory thus forces us to disambiguate our definition of supertask, and there is actually one natural way to do it. We can use Black’s idea -- presented in 2.4 -- of an infinity machine, a device capable of performing a supertask, to redefine a supertask as an infinite sequence of actions or operations carried out by an infinity machine in a finite interval of the machine’s own proper time measured within the reference system associated to the machine. This redefinition of the notion of supertask does not change anything that has been said until now; our whole discussion remains unaffected so long as ‘finite interval of time’ is read as ‘finite interval of the machine’s proper time’. This notion of supertask, disambiguated so as to accord with relativity theory, will be denoted by the expression ‘supertask-1’. Thus:

Supertask-1: an infinite sequence of actions or operations carried out by an infinity machine in a finite interval of the machine’s proper time.However we might also imagine a machine that carries out an infinite sequence of actions or operations in an infinite machine proper time, but that the entire process can be seen by an observer in a finite amount of the observer’s proper time.

It is convenient at this stage to introduce a contrasting notion:

Supertask-2: an infinite sequence of actions or operations carried out by a machine in a finite interval of an observer’s proper time.While we did not take relativity theory into account, the notions of supertask-1 and supertask-2 coincided. The duration of an interval of time between two given events is the same for all observers. However in relativistic spacetimes this is no longer so and the two notions of supertasks become distinct. Even though all supertasks-1 are also supertasks-2, there may in principle be supertasks-2 which are not supertasks-1. For instance, it could just so happen that there is a machine (not necessarily an infinity machine) which carries out an infinite number of actions in an interval of its own proper time of infinite duration, but in an interval of some observer’s proper time of finite duration. Such a machine would have performed a supertask-2 but not a supertask-1.

The distinction between supertasks-1 and supertasks-2 is certainly no relativistic hair-splitting. Why? Because those who hold that, while conceptually possible, supertasks are physically impossible (this seems to be the position adopted by Benacerraf and Putnam [1964], for instance) usually mean that supertasks-1 are physically impossible. But from this, it does not follow that supertasks-2 must also be physically impossible. Relativity theory thus adds a brand-new, exciting extra dimension to the challenge presented by supertasks. Earman and Norton (1996), who have studied this issue carefully, use the name ‘bifurcated supertasks’ to refer specifically to supertasks-2 which are not supertasks-1, and I will adopt this term.

Pitowsky (1990) first showed how this compatibility might arise. He considered a Minkowski spacetime, the spacetime of special relativity. He showed that an observer O* who can maintain a sufficient increase in his acceleration will find that only a finite amount of proper time elapses along his world-line in the course of the complete history of the universe, while other unaccelerated observers would find an infinite proper time elapsing on theirs.

Let us suppose that some machine M accomplishes a bifurcated supertask in such a way that the infinite sequence of actions involved happens in a finite interval of an observer O’s proper time. If we imagine such an observer at some event on his world-line, all those events from which he can retrieve information are in the ‘past light cone’ of the observer. That is, the observer can receive signals travelling at or less than the speed of light from any event in his past light cone. The philosophical problem posed by the bifurcated supertask accomplished by M has a particularly simple solution when the infinite sequence of actions carried out by M is fully contained within the past light cone of an event on observer O’s world-line. In such a case the relevant state of the world after the bifurcated supertask has been performed is M’s state, and this, in principle, can be specified, since O has causal access to it. Unfortunately, a situation of this type does not arise in the simple bifurcated supertask devised by Pitowsky (1990). In his supertask, while the accelerated observer O* will have a finite upper bound on the proper time elapsed on his world-line, there will be no event on his world-line from which he can look back and see an infinity of time elapsed along the world-line of some unaccelerated observer.

To find a spacetime in which the philosophical problem posed by bifurcated supertasks admits of the simple solution that has just been mentioned, we will move from the flat spacetime of special relativity to the curved spacetimes of general relativity. One type of spacetime in the latter class that admits of this simple solution has been dubbed Malament-Hogarth spacetime, from the names of the first scholars to use them (Hogarth [1992]). An example of such a spacetime is an electrically charged black hole (the Reissner-Nordstroem spacetime). A well known property of black holes is that, in the view of those who remain outside, unfortunates who fall in appear to freeze in time as they approach the event horizon of the black hole. Indeed those who remain outside could spend an infinite lifetime with the unfortunate who fell in frozen near the event horizon. If we just redescribe this process from the point of view of the observer who does fall in to the black hole, we discover that we have a bifurcated supertask. The observer falling in perceives no slowing down of time in his own processes. He sees himself reaching the event horizon quite quickly. But if he looks back at those who remain behind, he sees their processes sped up indefinitely. By the time he reaches the event horizon, those who remain outside will have completed infinite proper time on their world-lines. Of course, the cost is high. The observer who flings himself into a black hole will be torn apart by tidal forces and whatever remains after this would be unable to return to the world in which he started.

As Benacerraf and Putnam (1964) have observed, the acknowledgement that supertasks are possible has a profound influence on the philosophy of mathematics: the notion of truth (in arithmetic, say) would no longer be doubtful, in the sense of dependent on the particular axiomatisation used. The example mentioned earlier in connection with Goldbach’s conjecture can indeed be reproduced and generalised to all other mathematical statements involving numbers (although, depending on the complexity of the statement, we might need to use several infinity machines instead of just one), and so, consequently, supertasks will enable us to decide on the truth or falsity of any arithmetical statement; our conclusion will no longer depend on provability in some formal system or constructibility in a more or less strict intuitionistic sense. This conclusion seems to lead to a Platonist philosophy of mathematics.

Note, finally, the intuitionistic criticism of the possibility of supertasks is even less effective in the case of bifurcated supertasks, because in this latter case it is not required that there is any sort of device capable of carrying out an infinite number of actions or operations in a finite time (measured in the reference system associated to the device in question, which is the natural reference system to consider). In contrast, from the possibility of bifurcated supertasks in Malament-Hogarth space-times strong arguments follow against an intuitionistic philosophy of mathematics. As Earman and Norton remind us, it is noteworthy that certain facts relative to the non-Euclidean structure of space-time can have relevant consequences for the nature of mathematical truth.

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*First published: June 29, 1999*

*Content last modified: November 26, 2001*