If quantum-mechanical probabilities are most realistically thought of as single-case propensities (as the preceding chapters of this essay may indicate), the most realistic resolution of the following modern physical instantiation of Levy’s paradox (of equally likely integers) is that the natural numbers form a potential infinitude. Not an Intuitionistic potential infinity, but one of a more realistic kind (see option C-II in my 2005), one that bifurcated supertasks must, for their very possibility, simply assume is not the case (note that the set-theoretical axiom of infinity is hardly self-evident, and has little justification that is not sociological rather than metaphysical). My argument will be introduced via a pair of bifurcated supertasks (explained below), although the physical assumptions of my final instantiation of Levy’s paradox are more realistic than those that bifurcated supertasks require. For a tidier version of the final argument, see my 2008 Graduate Session in Aberdeen.
Levy’s paradox (attributed to Paul Levy in Cantelli 1935) can be explained quite simply. Suppose that you and I are selecting natural numbers blindly—e.g. by plucking tokens from bags containing À0 tokens (i.e. one per natural number)—in such a way that they are all equally likely to be picked. Whichever number you select, it is finite, so there are only finitely many smaller numbers that I might select, whereas there are infinitely many larger numbers. Since they are all equally likely, I am almost certain to select a larger number. But the same reasoning would apply to the number that I select, and so that you are almost certain to select the larger number, whence such a selection mechanism (one making the integers equally likely) is impossible.
Nonetheless, modern physics would seem to allow us such a mechanism, were it mathematically possible for physical objects to occur together in such numbers as À0. I shall first illustrate that claim via bifurcated supertasks, since their background complexities (which won’t be delved into) allow the mechanism to seen most clearly. A supertask is an endless sequence of physical tasks that has some final state (see Perez Laraudogoitia 2004 for an online introduction), and supertasks were traditionally used to challenge the coherence of the classical notion of infinity (i.e. the actual infinitude of the integers) by their physically implausible properties. Since the physics assumed by such supertasks was usually also classical (and therefore presumably false), such challenges were quite easily met (although not without questions arising as to the justification of the classical assumption of the actual infinitude of the integers, see my 2003).
Earman and Norton (1993 and 1996) recently turned the tables on that traditional use of supertasks, by using a relatively realistic sort of supertask, a bifurcated supertask (see below), to challenge the coherence of a non-classical notion of infinity, the Intuitionists’ potential infinity. Nonetheless bifurcated supertasks can also be used in the traditional way, to challenge the coherence of the classical assumption. I shall first show how a pair of such supertasks could display the inconsistency of Levy’s paradox, within an even more realistically quantum-mechanical universe. Now, that might just be due to the well-known incompatibility of such realistic ‘collapse’ interpretations of quantum mechanics and the relativity theory that allows bifurcated supertasks (as below), or to either alone. But I shall then argue that a similar paradox arises under very modest and plausible physical assumptions (as updated in the aforementioned talk of 2008). Finally I shall defend briefly (my 2006 arguments) the coherence of my preferred resolution of that paradox: that the integers are indefinitely extensible (for more of my 2006 thoughts, see here).
Earman and Norton (1996: 247-50) argue that bifurcated supertasks might be possible within Malament-Hogarth space-times. Such supertasks are called ‘bifurcated’ because, in such relativistic space-times, an agent in one region can have infinite time to carry out the supertask, while in another region an observer can observe the entirety of the supertask in a finite time.
Though such spacetimes are problematic in various ways, they are, we contend, not beyond the pale of physical possibility. If the Creator had a taste for the bizarre we might find that we are inhabiting one of them. (Earman and Norton 1996: 250)
Earman and Norton (1996: 250-6) considered in particular an infinity machine that could in principle (e.g. given À0 particles) spend forever running through all the natural numbers, testing any given number-theoretic conjecture against each number in turn, so that the observer (who might be a normal human) could know the truth or falsity of that conjecture, even were it a universal generalisation that happened to be true without any finite proof of it being possible within our finite language. That is how the classical notion of arithmetical truth might be secured against Intuitionistic scruples (e.g. such an infinity machine could be replaced by endless generations of Intuitionists, forming a single linguistic community to which the observer belongs).
Nonetheless it may well be, for all we really know, that any possible physical continuum (e.g. time, if time is continuous) must have a mathematical structure like C-II (see my 2005: 104). According to that relatively platonistic hypothesis, lines are full of points but do not contain transfinite numbers of points. So the possibility of bifurcated supertasks actually depends upon the infinitude of the natural numbers not being a certain kind of potential infinity, which is why such supertasks can also be used in the more traditional way, as follows.
Suppose that an observer can indeed, in a finite time, observe À0 successive tasks performed by a machine in an infinite time. Let each of those tasks be a mechanical test for whether or not a radioactive atom decays within its half-life, with each test being upon a different atom of the same isotope, but all the tests being as similar as possible. Each radioactive isotope does indeed seem to have a specific half-life—a duration for which the chance of such atoms decaying is exactly ½—associated with the tendency of such atoms to decay (when not subject to irradiation, etc.).
The decays of such atoms appear to be objectively chancy events, governed by the laws of quantum mechanics (in whichever formulation best fits reality), while the fact that single particles are associated with such chances is suggested by, for example, the famous two-slit experiments, whence I follow Popper (1983) in regarding such chances as single-case propensities. Many scientists (and some philosophers) regard such probabilities as frequencies—(for an online introduction to interpretations of probability, see Hajek 2003)—with such a probability of ½ being regarded as the limit, as the total number of half-lives tends to infinity, of a ratio of half-lives containing decays to some total number of half-lives. But intuitively, the observed frequencies provide, for each kind of particle, evidence for the underlying propensity that gives rise to them (see the preceding chapters). Propensities are epistemically obscure but they ought to be, being scientific posits, being noumenal rather than phenomenal (as frequencies are). Furthermore theories of propensity are works in progress, and if I am right then they ought to be based upon some rather novel mathematics anyway, so I shall keep my assumptions about quantum-mechanical chances as explicit and elementary as possible. Basically, I assume here finite additivity, finite multiplicativity, and (what is self-evident) that, given any scenario that (for any cardinal number v) has v exclusive and exhaustive possibilities, if a would be the case under each of those possibilities then the probability of a is 1. (For a tidier argument from 5 axiomatic properties, see my 2008 talk.) In particular I shall not assume that such chances are necessarily numerical, and so for clarity I shall reserve the word ‘probability’ (which is used in standard mathematics for a certain kind of real numerical measure, see section 1) for chances that are given by real numbers.
The machine doing the mechanical testing, then, is so set up that if the first n atoms decay (for some positive integer n), and the (n + 1)th does not, then it will send a message to the observer’s memory-banks to say that her integer is n, although if any further atoms decay (during their half-lives) then a second signal will be sent, to cause that first message to be deleted. So, if the observer can make arbitrarily precise observations, and store arbitrarily large numbers—(a big ‘if’ admittedly, but it will be dropped below (and note that Earman and Norton’s infinity machine also needed it)—then after a certain finite time she will have got, if anything, an integer.
Let Sn be the infinite sequence (n decays followed by nothing but non-decays) whose random occurrence causes the observer to get the integer n. Can anything be said about the chances of the Sn? Since the probability of an atom decaying during its half-life is by definition ½, hence the chance of getting any particular sequence of decays and non-decays could not be greater than infinitesimal because a sufficiently long finite sequence of decays and non-decays would, given finite multiplicativity for such probabilities, have a probability smaller than any given real number. Nonetheless some such sequence would be obtained by the testing-machine, and since both decay and non-decay are possible for each of the atoms in the endless sequence to be tested, hence it seems plausible that the Sn are possible in the same objectively random way that the atomic decays are possible.
Furthermore the Sn would all be equally likely to be received by the observer, as follows. Were they not, some sequence of half-lives, corresponding to some particular integer p, would be more likely to occur than some other sequence, corresponding to q. Were q bigger than p, those two sequences would differ in only q – p of the half-lives, and otherwise in only p – q of them. So p being more likely than q would have to be due to the difference in those half-lives, which are finite in number. But in each of those half-lives an atom is exactly as likely to decay as not, and so, via finite multiplicativity, p cannot be more likely than q after all. Note that the Sn need not have numerical chances. If one of them does have a numerical value (e.g. 0, or some other infinitesimal) then they would all have that same value, but to say that two possibilities are equally likely may just be to say that neither should be expected more than the other (and presumably any objective chances would be, to some extent, orderable), i.e. that the probability of either one, given that one of the two must occur, would be ½.
So consider two such observers, each observing their own machine, in this relativistic space-time, at some distance from each other. They are making a pair of observations that are intuitively simultaneous, i.e. such that each would see the other get the second integer were they both to get integers. Each observer knows that, whichever integer she gets (e.g. 1,000), only that many (1,000) of the positive integers that the other observer might then get are not bigger than it, while more than ninety-nine times as many (>99,000) of them are bigger—and because each of them also knows that the positive integers are all equally likely, each observer reasons that the other observer is very likely (>99%) to get a bigger integer.
Each observer so reasons; but of course, they cannot both be right. In fact, by the symmetry of the situation it would seem that, were they both to get integers, neither would be more likely to get the bigger one. Of course, it is extremely unlikely that they would both get integers, but a problem arises just because it is possible that they would. And of course, two people may well come, in reasonable ways, by conflicting expectations about even real-world (and therefore presumably coherent) scenarios, but the underlying problem here does not appear to be due to such subjectivity (see §4 and §5 below). To begin with, those two observers occupy a relativistic space-time, so that their vantage points (from which each would see the other getting the second integer) are as objective as any could be. And furthermore, they were both reasoning soundly about the objective situation, just as we would have to do to consider the objective chances. So in order to examine their reasoning in more detail, let us now consider the various possibilities, for one of the observers.
She might get ‘1’, and then surely we (from our objective viewpoint) ought not to think that the other observer would actually have, at that point in this scenario, an objective probability of getting, if anything, a ‘1’, that was as big as ½, and a probability as small as ½ of getting, if anything, one of the other integers. To labour the point a little, note that while ½ would be the correct value if the other observer could only get, if anything, ‘1’ or ‘2’, in fact she could also get the equally likely ‘3’, or ‘4’, ..., or ‘99’, or ‘100’ (etc.). Even if those hundred integers were the only possibilities, the probability of her getting ‘1’ would be 1%. And although there are more possible values, it is surely implausible that those extra possibilities would raise that chance.
And while it could be that a hundred equally likely possibilities would be no more likely than just one of them, that would only be the case if their probabilities were all 0, in which case the probability of getting ‘1’ would certainly be less than 1% (via finite additivity). More generally, if it does make sense to assign numerical values to the conditional chances of the Sn (i.e. to the chances of the Sn given that the observer gets something) then those values could not be bigger than infinitesimal (since if they were non-infinitesimals then adding together some finite number of them could yield a value bigger than 1), whence the chance of getting ‘1’ in particular would be nowhere near as big as 1%.
Now, she could get ‘2’ instead, but it is just as implausible that the other observer would then have an objective chance greater than 1% of getting, if anything, ‘1’ or ‘2’, there being at least 198 other integers that she could get instead; and a plausible objective chance is again infinitesimal, given finite additivity. She could get ‘3’, but similarly there are at least 297 other integers; and so forth. In short, whichever integer our observer gets it would seem that the objective chance that the other observer would then get the bigger integer could not be anything like as small as 99%, let alone ½.
So, were such pairs of observations made repeatedly, the other observer would probably tend to get the bigger integer, in those cases where both observers get integers. But of course, we can similarly consider this scenario (in the same objective way) from the other observer’s position, and so there really does seem to be a conflict: how could it be very likely that they would both tend to get the bigger integer? And if their reasoning is not at fault then it must be the scenario that is impossible.
Since the arbitrarily precise observations and accurate storage of arbitrarily large numbers that were required were rather implausible (see Earman and Norton 1993), the following scenario is set more plausibly in this universe (and contains no subjective elements). To begin with, it appears that time is endless. E.g. the endlessness of time is suggested by recent astronomical observations, which indicate that the rate of expansion of the universe has been increasing. But maybe time is not endless, so I shall consider other possibilities too (see below). Secondly, although there seems to be only a finite number of particles at present, empirical observations (e.g. those of the Casimir effect) also indicate that such particles exist within a vacuum that fluctuates randomly as virtual particles appear spontaneously and almost immediately annihilate one another.
[We may think] of vacuum fluctuations as pairs of virtual particles that appear together at some point of spacetime, move apart, and come back together and annihilate each other. ‘Virtual’ means that those particles cannot be observed directly, but their indirect effects can be measured, and they agree with theoretical predictions to a remarkable degree of accuracy. (Hawking 2001: 118)
So even so-called empty regions of space may be associated with some objective probability of (some number of some kinds of) virtual particles appearing during some interval of time. The probability of such particles appearing would clearly be higher for larger regions of space-time (and smaller numbers, fewer kinds, and so forth) and lower for smaller regions (bigger numbers, more kinds). So if space-time is a continuum then we could consider, in place of the half-lives used above, spatio-temporal regions for which the probability of (some number of some kinds of) virtual particles appearing in that spatial region, during that time, is ½. I shall call such regions ‘Binary Regions.’
Were Binary Regions objective parts of space-time, the endlessness of time would imply that À0 Binary Regions would exist in this universe. But even if time is not endless, and this universe collapses back upon itself after all, it is not especially unlikely that there would then be another Big Bang, with that cycle repeating either endlessly or else until an eternal universe occurred, so that we could still consider À0 Binary Regions or even, in the former case, À0 half-lives. And whatever happens to this universe, there might be an infinite number of other physical universes, within a multiverse (see Rees 2001). And for all we know, the particles in this universe might be refreshed via white holes (or via some other mechanism). In either of those two cases we could also consider À0 half-lives (not in this universe in the former case, but objectively enough nonetheless). Furthermore, some recent theories allow this universe to be spatially infinite, in which case we could again consider À0 Binary Regions, and maybe À0 half-lives too. And of course, there may well be other physical processes with two equally likely outcomes, which could occur in À0 instances in this universe (or multiverse).
So for the sake of generality, I shall call the two equally likely outcomes (of a Binary Region, or of an atom’s half-life, etc.) ‘Heads’ and ‘Tails’, and any such process a ‘Toss’. Note that such Tosses can be causally independent of each other, as follows. Clearly they could be were there multiple universes, because we could consider just one Toss per universe. And such independence would only be a matter of stipulation for Binary Regions in an eternal universe—see my 2008 handout for a nice analogy for this—because the size of the spatio-temporal region that is any particular Binary Region is simply whatever gives a probability of ½ in that particular case; and similarly for the particular half-lives of individual atoms (such half-lives might vary slightly with local conditions, but the probabilities of interest are objective features of the world, not values that need to be ascertained by some observer). So, some of the combinatorially possible outcomes of an endless sequence of Tosses begin with a finite number of Heads, followed by an endless sequence of Tails, and such outcomes will be called ‘Integers.’ E.g. the Integer n has n Heads followed by Tails endlessly.
The fact that Integers are possible follows from the fact that for each Toss there is some objective probability of getting a Tail, so that, because those probabilities are single-case propensities, an endless sequence of Tails is possible. Furthermore, since those probabilities are all ½, the Integers are equally likely outcomes of an endless sequence of Tosses (for such reasons as were given above). Now, it may seem to some readers that if Heads and Tails were equally likely (as with the fair toss of a fair coin) then it would not be possible to get an endless sequence of Tails (since such an outcome would show that the coin was biased). Since such readers would, I think, be confusing epistemology (why the coin would be called ‘fair’) with metaphysics (whether or not the coin is fair), it may help them to consider *Integers* as follows. Let z be any endless sequence of Heads and Tails that would be regarded by such a reader as possible (and note that there must be some such z, unless À0 ought really to have been a potential infinity). Presumably z will contain À0 Heads and À0 Tails, distributed (randomly) in such a way that their limit frequencies are both ½. Let the *Integer* n be obtained from the sequence z by replacing its first n elements, Heads for Tails and Tails for Heads. Since z appears possible, each of the *Integers* should also appear possible (and they are again all equally likely).
Anyway, since two spatial regions or two atoms could be considered at a time (or the odd and even elements of one sequence could be considered separately, etc.), we may clearly consider two endless sequences of Tosses. And it is quite possible (if highly unlikely) that both sequences would yield Integers. One well-defined subset of the set of all the possible outcomes is the set of pairs of unequal Integers (pairs of equal Integers being excluded for the sake of a tidier argument here), which I call ‘P’. Can much be said about the conditional probabilities within P, the probabilities given both that both sequences yield Integers and that those two Integers are different? To find out, let two Integer variables (corresponding to the two endless sequences of Tosses) be n and m. Within P, either
(X) n will be smaller than m, or else
(Y) m will be smaller than n, exclusively and exhaustively.
Recall that n may be any positive integer (so long as it does not equal m), and that similarly m may also be any positive integer (so long as it does not equal n). Since n and m are obtained in exactly the same kind of way, hence it will generally be the case that whatever can be said of n can also be said of m (with m in those statements replaced by n), whence it is plausible that neither X nor Y could be more likely than the other to occur, simply by the symmetry of the situation. And since either X occurs, or else Y does, exclusively and exhaustively, it is therefore plausible that the conditional probability of Y (in particular) occurring is ½.
Can anything else be said, about the conditional probability of Y? Since that is just the probability within P of n taking some value, and m taking some value, both such that m < n, it seems plausible that, n and m being mathematically independent of each other (all the Tosses being causally independent of each other, and all the chances being single-case propensities), we may consider n first and then m (or indeed, m first and then n). If so then, whatever the value of n, either
(An) m will be one of the first 100·n – 99 positive integers, or else
(Bn) it will not.
Note that if Bn is the case, then m is bigger than n, and that within An there are only n – 1 positive integers smaller than n, while 99·(n – 1) are bigger than n. So, whatever the value of n, at least 99 times as many of the equally likely values of m are such as give rise to X, rather than Y.
If Bn could not occur then, whatever the value of n, the conditional probability of Y would be either 1% or (were the conditional probabilities of the m all 0) 0. Of course, then m and n would not be independent; but still, it is implausible that the extra possibilities in Bn, which would all give rise to X, would make that value bigger than 1%.
Furthermore, in a couple of cases we can be more precise, because only finitely many values of m give rise to Y, given any n. If the conditional probabilities of the various values of m were all 0 then, whatever the value of n, the conditional probability of Y given that value of n would also equal 0, via finite additivity. Similarly, were those conditional chances (of the values of m) some sort of nonzero infinitesimal then the conditional chance of Y given some value of n would also be no bigger than infinitesimal, via finite additivity, and hence again smaller than 1%.
So, since the conditional probability of n taking some value is (within P) 1, and since the conditional chance of m then taking a value such that Y is (according to the plausible arguments above) no bigger than 1%, hence it is plausible that the conditional chance of Y is not a number bigger than 1%. Note that even though that last step takes us into contradiction with our previous result (that the conditional probability of Y occurring is plausibly ½), it can be justified as follows. Let v be any cardinal number, and suppose any scenario with v exclusive and exhaustive possibilities. Clearly, if each of those possibilities yields a then (whatever the outcome of the scenario) a is certain; the probability of a is 1. The last step above is justified because we may let a be the conditional chance of Y not being bigger than 1%, and let v = À0. Now, although that certainty of a (given those v possibilities) is not normally regarded as an elementary property of probabilities, note that there are varying degrees of consensus about which properties should be taken to be elementary (e.g. see Humphreys’ paradox), and that at least that property is self-evident. E.g., consider a box containing only v apples, for unknown v. If an object is picked out of that box, then clearly it must be an apple. Clearly that probability of 1 does not depend upon anything else; not upon which cardinal v is, nor upon the probabilities being completely additive numbers (countable additivity if v = À0).
Incidentally, the reader may well be aware of a different approach, which is mathematically superior in well-behaved scenarios; i.e., if the chance of a occurring via one of those v possibilities is, for each one, 1/v, then the chance of a is the mathematical sum of all those v chances. When v is finite, that approach gives the probability of a as v/v = 1, as desired. But in order to take that approach with v = À0 we would need numerical chances of 1/À0 and an axiom of countable additivity, in order to get the result that a will be the case. And a requirement for those two would mean that the step in question (before which neither or those two had been assumed, let alone justified) was unjustified. Nonetheless I need not assume that sense can be made of 1/À0 here, and nor need I assume countable additivity. After all, while this alternative (more standard) approach may well be more convenient if we are looking at consistent situations described by a convenient mathematics, even it would ultimately have to be justified via such self-evident properties as the one that I am assuming (and which applies directly to the underlying possibilities and certainties).
Philosophical arguments all boil down eventually to the relative plausibility of conflicting intuitions; and in my opinion, the scientific plausibility of my assumptions and the intuitive validity of my argument imply that the objective conditional probability of Y is indeed contradictory. But of course, some philosophers will regard such conclusions as so undesirable (e.g. if they like single-case propensities and standard set theory) that they will imagine that my argument must have been fallacious. So, to what intuitions could such philosophers appeal? One suggestion has been that I consider the following picture.
Since m and n are independent variables, it is natural to put them on orthogonal (Cartesian) x and y-axes respectively, so that we are considering the bottom left-hand corner of an infinite square. The leading diagonal represents m = n, with Y (m < n) above it and X (m > n) below, and although the diagonal does not actually cut this infinite square in half (not in the way that it would cut a finite square in half), we may surely equate the probability of being above the diagonal, in this infinite case, with the limit, as s goes to infinity, of the probabilities of being above the diagonals of finite squares for which m and n are no greater than s, which gives the probability of Y as ½. Furthermore the horizontal lines of (m, n) values correspond to our having got one value for n (vertically) and being then able to get any value for m (the horizontal line). For any such line (i.e. for any value of n) there are infinitely many m-values in X and only finitely many in Y, but for finite squares there is more symmetry. For finite squares, the ratio of m-values in X to m-values in Y varies with n, whence it makes sense to average them out, i.e. to sum over the n and divide by s. And taking the limit gives the probability of Y as ½ (and similarly for vertical lines).
Nonetheless that picture only encourages us to make a fallacious analogy with the finite (if no more so than supertasks often do), because we cannot always equate the values for infinite systems with the limits of similar finite systems. E.g., adding v objects to a box and then taking them away again leaves v – v = 0 objects in the box if v is finite, yet if v = À0 then any number of objects, between 0 and À0 inclusive, might be left behind. That picture explains nothing about the original problem (of §3), that whichever integer our observer happened to get, the chance of the other observer then getting the bigger integer could hardly be as small as 99%, let alone ½.
Such philosophers might instead appeal to our intuitions about the consistency of real analysis, by pointing out that, given countable additivity, there is no uniform probability distribution on the integers. But while that is indeed the case (and while real analysis may well be consistent) it too does nothing to explain the original problem, or to stop the endless sequences of binary regions (or decaying atoms, etc.) from being plausible. Given countable additivity, and that my chances are probabilities, we would indeed have a shorter argument to the same paradoxical conclusion, but the point of the argument was that it was independent of such dubious assumptions. So consider how we might prevent the problem arising.
A first thought might be that the problem arises because of false physical assumptions (as tends to be the case with paradoxical supertasks). Perhaps there are in fact no Tosses, or perhaps, while there is some finite number of causally independent Tosses, there are not À0 of them. Regarding the former option, there are ‘no-collapse’ interpretations of quantum mechanics, and I shall not argue against them here. I just think that we ought to fall back on them (as we would fall back upon Idealism) only when we have to. The question here is, do we really have to? The latter option (of allowing some Tosses, but not, for physical rather than mathematical reasons, À0 Tosses) is clearly undesirable because it would have us conclude, in the face of such empirical evidence, that the universe will collapse back upon itself, to explode again at most only a finite number of times (in order that this universe will not contain À0 Binary Regions). Furthermore we should also have to outlaw such possibilities as À0 universes, which would undermine, for example, the strong anthropic explanation of our universe’s convenient physical constants (see Rees 2001).
Perhaps the reason why those options seem disproportionate (if we have not already favoured them) is that the conclusion of my argument was so clearly a consequence of the abstract structure of the À0 Tosses; so, perhaps we have false mathematical assumptions. (Some may find it odd that physical thought-experiments would be able to tell us about the nature of number rather than matter, but we may of course reject a mathematical model for generating nonsense.) Could it be that such Tosses, at least when there are À0 of them, are not independent of each other? But it is quite obscure how that could be the case for an actual infinity of single-case propensities. Suppose that there were À0 causally independent universes; how could they fail to be mathematically independent? Similarly, it would be unrealistic to interpret my argument as an argument for a frequency interpretation of quantum-mechanical probability. And because of the popularity of set theory, the possibility that my argument is for the infinitude of the integers being potential (in a realistic way) rather than actual will also seem disproportionate.
Nonetheless it should not. Such a resolution would also resolve the Banach-Tarski paradox, for example, which arises within the measure-theoretic foundations of standard probability theory (incidentally both Lebesgue, who founded modern measure theory, and Kolmogorov, who founded modern probability theory, were Intuitionists of some sort), so prima facie such a resolution would (were it more realistic than the Intuitionism refuted by Earman and Norton) be in the right ballpark. (After all, there is surely something odd about infinite space containing exactly À0 unit volumes and yet having no measure.) Note that having a potentially infinite totality of natural numbers could be compatible with there being an actually infinite totality of universes (e.g. there might be # of them, where the cardinal # = 1/0 was introduced in my 2005), an actually infinite space-time, etc., so that such a resolution might avoid the problems (mentioned above) associated with the outlawing of À0 Tosses for physical (rather than mathematical) reasons.
Certainly, either the natural numbers form an actual infinitude (in the standard sense) or else they do not, they are instead some sort of potential infinity. The latter tends to be ignored nowadays, but is there a good reason for assuming the former within metaphysics? Of course, science tends to be written in the language of ZFC set theory nowadays (if only implicitly), but although that is indeed a pragmatic reason for us to assume, in our everyday practices, the former, it is hardly an argument for the logical impossibility of the latter. It does indicate that there should be some such argument, somewhere, but where? Still, many of us feel a natural aversion to semantic anti-realism about arithmetic, so it would be unsurprising were such forms of Intuitionism refutable, but even that would only be a reason for ignoring the latter were there no way in which it could be conceived of more realistically, whereas there is (see below, and here). Now, contradictions within ZFC are certainly elusive (if they exist at all), but consistency is hardly sufficient for a correspondence with reality (e.g. Ptolemaic astrophysics was consistent). And what about the elusiveness of the fundamental intuitions that would make actual infinities out of potential infinities without also making contradictions out of the proper class of actual infinities? Were the success of ZFC a reason for ignoring the potential infinity possibility, the latter elusiveness would seem to be a reason for not ignoring it completely.
Both possibilities can probably be modelled well enough by logical systems, and they both have fairly common intuitions in favour of them. So, just as we might counter Intuitionistic claims, about the incoherence of the actual infinity, by realistically positing such a possibility, so too the intuitions that favour the actual infinity are not, by themselves, decisive. There is, for example, the ease with which we can conceive of À0 objects within an infinite space-time (e.g. one per year, or per light-year). But were those realistic objects, they would presumably be quantum-mechanical (and the above indicates that À0 such objects are not that easy to conceive of), whereas were they not then we would hardly have much of a reason (ease of conception being no guarantee of possibility, as many paradoxes show).
Of course prima facie it may well seem that, since we may subdivide continua endlessly (at least conceptually), they must already contain À0 parts, À0 points if (as I shall assume for simplicity) continua are full of (some actual infinity of) points. However, although within a line of points there will indeed be a first point, a second point, a third and so forth, to think of such points is only to think of the start of an endless process, which might (in the first place) be either actually or potentially infinite. It is true that since each product of that process is already a point in the line, hence all the products are in the line already. But it only follows that there are À0 points in the line if that endless process may be taken to its completion (so that the totality of its products would be an actual rather than a potential infinitude) and of course, taking that process to its completion requires that there are À0 points. While it is natural for us to think that there would have to be À0 objects in any actually infinite collection of objects, surely it is only our experience with finite collections leading us to expect that. Russell’s paradox indicated—and still indicates (see Fletcher 2007: 531-51)—that we have no firm grasp of anything else.
The intuition that there would be À0 points in a line full of points is therefore akin to the intuitions utilised unsuccessfully by the traditional supertasks to indicate the impossibility of À0 objects. Consider a classical planet beginning to orbit an eternal sun forever. Every single one of its orbits will only be a finite number of orbits away from the first orbit, and yet there are infinitely many orbits, so (to put it rather crudely) every single orbit is in the first 0% (to the nearest real ratio) of the collection of all of them. Similarly, think of the finite cardinals lying before you, as a line of tokens (coexisting spatially), and then ask yourself how big they get. There is no gap between them and infinity, because they are all of the finite cardinals, and yet because they are only the finite cardinals it is hard to see how they could go all the way up to infinity. Still, such thoughts as those two no more show the potential infinitude of the integers than the previous argument showed their actual infinitude. That the endless sequence of integers is potentially infinite, and that it is actually infinite are both, so far as we know, logically possible, and neither is prima facie implausible (see below) although both are counter-intuitive. But only one will best describe the structure of reality, whence we might expect that the assumption of the other would yield paradoxes (as above).
I shall end by saying a bit more about my 2005. To begin with, if there are physical continua (e.g. space-time) then it is not especially implausible that they would be full of physical points (a transfinite cardinal number of them perhaps, or # of them). We could hardly construct our primitive concept of a nonzero-dimensional continuum using only the primitive concepts of points and pluralities, so it is natural to think of lines (and planes and spaces) as being, in the first place, nonzero-dimensional continua, only then considering them to be totalities of points, to be full of points (rather than made of points), e.g. by considering that a point might be anywhere within them, there being nowhere where something of size 0 could not fit. So it is not implausible that the actual infinity that is the number of points in a line of points should be obtained directly in a single step (e.g. via the power-set operation perhaps, or by a division by 0), i.e. in a non-hierarchical way. On the other hand, the natural numbers are 1, and 2 = 1 + 1, and 3 = 2 + 1, and so forth (continuing to add 1 in that way), and so their totality, being just all of those numbers, is clearly an infinitude that is, at least in the first place, obtained in an endlessly hierarchical way. Must a non-hierarchically infinite collection (such as a line of points) contain a substructure that is an endlessly hierarchical totality (such as À0 points)? It is quite plausible that, regarding two such fundamentally different kinds of infinite collection, we simply do not have clear enough intuitions (ones that do not make uncertain analogies with the finite) to say how they are likely to be related, which is why it is not prima facie implausible that continua, whilst being full of points, do not contain À0 points.
Incidentally, section 10 was inspired by Brown’s (2004: 38-9) elucidation of Freiling’s 1986 argument against the axiom of choice: Suppose that the real numbers between 0 and 1 can be obtained randomly, so that each is equally likely to be obtained (e.g. via an w-sequence of coin-tosses, with each combinatorially possible sequence of heads and tails corresponding to an endless binary expansion). Given the axiom of choice, the reals can be well ordered, with each real number dividing that ordering into an initial part (of measure 0) between the first reals and that one, and a final part (of measure 1) containing the rest of the reals (much as each natural number divides the w-sequence 1, 2, 3, …, into an initial finite part and an infinite remainder). Two such random reals will therefore behave paradoxically (since intuitively the other real has not a 0% but a 50% chance of being in the measure-0 part; for the details, see Freiling 1986); and since the axiom of choice (or something very like it) seems to be an intrinsic part of our conception of an actual infinitude (of actual things, since it merely says that, given a set of so many things, there are also all the subsets of those things), hence Freiling’s argument is half-way to being an argument against their actual infinitude.
For
their comments on this supertask (which is still changing, post-talk and
pre-publication) many thanks to Michael Butler, Kit Fine, Peter Fletcher, David
Fremlin, Wilfrid Hodges,
John Norton, Philip Percival, Alexander Pruss, Adam
Rieger, Eric Schechter, Alan Weir, Christopher Yorke and all
those whose names I’ve forgotten (those who know me know that I can’t remember
names for toffee).