THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 12 SURREAL TIME AND ULTRATASKS HAIDAR AL-DHALIMY Department of Philosophy, University of Minnesota and CHARLES J. GEYER School of Statistics, University of Minnesota Abstract. This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. §1. Introduction. To perform a hypertask is to complete an uncountable sequence of tasks in a finite length of time. Each task is taken to require a nonzero interval of time, with the intervals being pairwise disjoint. Clark & Read (1984) claim to “show that no hypertask can be performed” (p. 387). What they in fact show is the impossibility of a situation in which both (i) a hypertask is performed and (ii) time has the structure of the real number system as conceived of in mainstream mathematics—or, as we will say, time is R-like. However, they make the argument overly complicated. THEOREM 1.1. It is not possible, in R-like time, to perform a hypertask. Proof. Let t be the finite amount of time it takes to accomplish all of the tasks. For each integer n,atmostnt of them can take time greater than 1/n. All tasks take some nonzero time. Thus, the set of all tasks is a countable union of finite sets, which is countable. This kind of argument is familiar from many areas of mathematics. Clark & Read cite Cantor without stating how simple the argument really is. Theorem 1.1 relies on the properties of the real numbers. If one uses the surreal num- bers (Conway, 2001, first edition 1976) instead, then it is possible to perform not only a hypertask, but also what we will call an ultratask,1 which includes the doing of one task for each ordinal number.2 Ultratasks are a kind of hypertask, since there are uncountably many ordinals. We will call time that is structured like the surreal numbers surreal time. There are many different notions of possibility: there are various kinds of logical possibility (corresponding to different logics), various kinds of mathematical possibility Received: May 13, 2016. 1 Ultratasks have no relation to ultrafilters or ultraproducts. Szabó (2010) defines “ultratask” differently. 2 Thanks to Roy Cook for the idea of an ultratask. c Association for Symbolic Logic, 2016 1 doi:10.1017/S1755020316000289 2 HAIDAR AL-DHALIMY AND CHARLES J. GEYER (classical ones, constructive ones, intuitionistic ones, and others), and there are conceptual, epistemic, nomic, physical,andmetaphysical notions of possibility. Our main claim is that surreal time and hypertasks are mathematically possible, in the sense of being compatible with the truths of classical mathematics.3 This kind of mathematical possibility will be indicated by the subscript cm.4 THEOREM 1.2. It is possiblecm, in surreal time, to perform an ultratask. Proof. For each ordinal α, start the α-th task at time α/(α + 1). Readers who are not familiar with the surreal numbers may not understand this proof. We explain the surreal numbers in §2 and §4, and expand and discuss the proof in §3. §2. Surreal numbers. The surreal number system No (Conway, 2001) is a totally ordered Field with a capital “F” (Conway, 2001, chap. 1); Conway capitalizes names for mathematical structures whose domains are proper classes. No includes the real number system R as a subfield, but is much larger. No includes (as a subclass) the proper class On of all ordinals. The real numbers and ordinal numbers, considered as elements of No,have all of the properties of the real numbers and ordinal numbers from ordinary mathematics (Conway, 2001, chap. 2). No is also a real-closed Field (Conway, 2001, chap. 4), meaning No has all the first-order properties that R has. Hence, if α is a nonzero√ surreal number, then so is 1/α,andifα is a non-negative surreal number, then so is α. Any arithmetic and algebraic operations that take real numbers to real numbers also take surreal numbers√ to surreal numbers. In particular, if α is a nonzero ordinal, then α − 1, 1/α,and α are surreal numbers. The proof of Theorem 1.1 would not work if time can take values in the surreal numbers. It would go wrong because it assumes non-first-order properties of R that No does not share. In particular, it assumes that the sequence 1/nn∈N converges to zero—although this holds in the real numbers, in the surreal numbers this sequence does not converge to zero. There are many infinitesimal surreal numbers smaller than 1/n for all integers n,for example, 1/α for any infinite ordinal α. Formally, it is not true in No that for every ε>0 there is a natural number N such that 1/n <εfor all natural numbers n ≥ N. It is not true whenever ε is infinitesimal—this, in fact, is the meaning of the word ‘infinitesimal’. §3. Ultratasks. Recall that an ultratask includes the doing of one task Tα for each ordinal α. If time takes values in the surreal numbers, then there is an easy way to do an ultratask in one unit of time.5 Define the function f by x f (x) = , x ≥ 0. (1) x + 1 This function makes sense for surreal numbers x just like for real numbers x. We could replace f by any other algebraic function taking non-negative argument values that is strictly increasing and bounded above by a real number. Start Tα at time f (α). 3 It suffices here to include NBG with Global Choice among “classical” mathematics. 4 We take it that whatever is possiblecm is also logically possible (in the sense of classical logic), but not vice versa. 5 In fact, an ultratask—or even a separate ultratask for each ordinal—could be performed in an arbitrarily short positive interval of surreal time. SURREAL TIME AND ULTRATASKS 3 The length of time between the start of Tα and the start of Tα+1 is α + 1 α (α + 1)2 − α(α + 2) 1 − = = . (2) α + 2 α + 1 (α + 1)(α + 2) (α + 1)(α + 2) The operations here are the Field operations for No, which do not agree with the usual ordinal arithmetic used in set theory (Conway, 2001, p. 28). The addition and multiplica- tion operations for No when applied to ordinals are called the natural or Hessenberg or Hessenberg–Conway sum and product in set theory. There are no subtraction or division operations defined for ordinals except the Conway ones, which are the Field operations for No. If α is an infinite ordinal, then the length of time (2) is infinitesimal, but it is a strictly positive surreal number. If one likes rests (staccato performance, Clark & Read, 1984, their §3), then one may (optionally) take part of this interval to do Tα and the remainder of the interval to rest. As we shall see in §6, there may be rests even if one does not have rests of the kind just described. §4. Dedekind cuts. Is the theory in §3 philosophically satisfying? Is it philosophically legitimate to consider surreal time? Does any philosophical principle dictate that time is necessarily R-like, in which case, by Theorem 1.1, no hypertask would be doable in any possible world in which there is time? The real numbers can be constructed as Dedekind cuts of the rational numbers. Dedekind cuts were indeed a great idea (Reck, 2012), but if they are such a great idea, why stop at the rationals? What about Dedekind cuts of the real numbers? Why aren’t they a great idea too? We raise this issue because it leads to a construction of the surreal numbers. We quote Conway (2001, pp. 3–4): Let us see how those who were good at constructing numbers have approached this problem in the past. Dedekind (and before him the author—thought to be Eudoxus—of the fifth book of Euclid) constructed the real numbers from the rationals. His method was to divide the rationals into two sets L and R in such a way that no number of L was greater than any number of R, and use this “section” to define a new number {L|R} in the case that neither L nor R had an extremal point. His method produces a logically sound collection of real numbers (if we ignore some objections on the grounds of ineffectivity, etc.), but has been criticised on several counts. Perhaps the most important is that the rationals are supposed to have been already constructed in some other way, and yet are “reconstructed” as certain real numbers. The distinction between the “old” and “new” rationals seems artificial but essential. Cantor constructed the infinite ordinal numbers. Supposing the inte- gers 1, 2, 3, ... given, he observed that their order-type ω was a new (and infinite) number greater than all of them. Then the order-type of {1, 2, 3,...,ω} is a still greater number ω+1, and so on, and on, and on.