Infinite lotteries, spinners, and the applicability of hyperreals
aa r X i v : . [ m a t h . HO ] A ug INFINITE LOTTERIES, SPINNERS, AND THEAPPLICABILITY OF HYPERREALS
EMANUELE BOTTAZZI AND MIKHAIL G. KATZ
Abstract.
We analyze recent criticisms of the use of hyperrealprobabilities as expressed by Pruss, Easwaran, Parker, and William-son. We show that the alleged arbitrariness of hyperreal fields canbe avoided by working in the Kanovei–Shelah model or in saturatedmodels. We argue that some of the objections to hyperreal prob-abilities arise from hidden biases that favor Archimedean models.We discuss the advantage of the hyperreals over transferless fieldswith infinitesimals. In [18] we analyze two underdetermination the-orems by Pruss and show that they hinge upon parasitic externalhyperreal-valued measures, whereas internal hyperfinite measuresare not underdetermined.
Contents
1. Introduction 22. On mathematical representation of physical processes 52.1. Are hyperreal fields arbitrary? 62.2. Pruss admits failure of arbitrariness/ineffability claims 62.3. Shift-invariance hypothesis 72.4. Cardinality objection 92.5. Standard model of the naturals and coinflipping 103. On the strength of theories with infinitesimals 113.1. Constructing hyperreal fields 123.2. The transfer principle of Robinson’s framework 123.3. Usefulness of infinitesimals: Klein and Fraenkel 143.4. Infinitesimals without transfer 164. Conclusion 18Acknowledgments 20References 20
Date : August 27, 2020.2010
Mathematics Subject Classification.
Primary 03H05; Secondary 03H10,00A30, 60A05, 26E30, 01A65.
Key words and phrases.
Infinitesimals; hyperreals; hyperfinite measures; internalentities; probability; regularity; axiom of choice; saturated models; underdetermi-nation; non-Archimedean fields. Introduction
Since Abraham Robinson introduced his framework for infinitesi-mal analysis in the 1960s (see [68] and [69]), a sizeable literaturehas developed in connection with the applicability of said infinitesi-mal analysis in probability theory, physics, and other fields. Despitethe growing body of literature featuring such applications, the re-cent years have seen a vigorous debate concerning the applicability ofRobinson’s framework in the sciences, with a number of advocates andalso a number of critics. The latter include Easwaran, Elga, Parker,Pruss, Towsner, and Williamson. Recent additions to the literature areEaswaran–Towsner ([28], 2018), Pruss ([65], 2018), and Parker ([62],2019). Easwaran and Towsner call into question the applicability ofRobinson’s framework to the description of physical phenomena. Arebuttal appears in Bottazzi et al. ([17], 2019). The present articlefocuses mainly on the critiques as formulated by Parker, Pruss, andWilliamson. These authors have questioned the applicability of hyper-real models in probability.In the present text and in the sequel article [18], we analyze a claimby Alexander Pruss (AP) that hyperreal models are underdetermined ,in the sense that, given a model, allegedly “there is no rational reason tochoose a particular infinitesimal member of an extension to be a valuefor the probability” ([65, Section 3.1]) of a single event. To buttress hisclaim, AP exhibits measures assigning a different infinitesimal value tothe event. We argue, however, that all of AP’s additional measuresare parasitic in the sense of Clendinnen [21]. More specifically, APignores a key property of entities such as functions and measures inRobinson’s framework, namely the property of being internal . Theimportance of internality stems from the fact that Robinson’s transferprinciple only applies to internal entities. Meanwhile, we prove that all Applications to physics (Albeverio et al. ([1], 1986), Faris ([31], 2006), Van denBerg and Neves ([81], 2007), Loeb and Wolff ([57], 2015)), to probability theory(Nelson [61], 1987), to stochastic analysis (Capi´nski and Cutland [20], 1995), tocanards (Diener and Diener [24], 1995), to mathematical economics and theoreticalecology (Campillo and Lobry [19], 2012), to error analysis (Dinis and van den Berg[25], 2019), to Markov processes (Duanmu et al. [26], 2021). Clendinnen points out the possibility that “All members of any set of empiricallyequivalent but logically distinct theories might be parasitic on one key theory. Thatis each of the other members of the set might only be able to be formulated byutilizing the formulation of the key theory. If this situation obtained, [differentialunderdetermination] would not hold; for the predictions which could be made byusing any one of the set of theories would nevertheless require the selection of thesingle key theory. So the making of these predictions would depend on the selectionof a unique theory” [21, p. 76].
NFINITE LOTTERIES, SPINNERS, APPLICABILITY OF HYPERREALS 3 of AP’s additional measures are external . Thus, if one considers onlyinternal hyperfinite measures, no underdetermination occurs.AP also claims that certain transferless ordered fields properly ex-tending the reals, such as the Levi-Civita field or the surreal numbers,may have advantages over hyperreal fields in probabilistic modeling.However, we show in [18] that probabilities developed over such fieldsare less expressive than real-valued probabilities, and inferior to prob-abilities developed over hyperreal fields.In more detail, AP ([65], 2018) attacks Robinson’s framework formathematics with infinitesimals, claiming that its applications in prob-ability cannot have any physical meaning. AP’s critique is more sophis-ticated than that of Easwaran–Towsner, in that he acknowledges at theoutset that some commonly voiced objections to hyperreal numbers areunconvincing. Nevertheless, AP claims that hyperreal probabilities are underdetermined , namely that there is no rational reason to assign aparticular infinitesimal probability to non-empty events that classicallyhave probability zero. His argument hinges upon the following: • examples of uniform processes that allegedly do not allow for auniquely defined infinitesimal probability for singletons, and • a pair of theorems asserting that for every hyperreal-valuedprobability measure there exist uncountably many others thatinduce the same decision-theoretic preferences.We will argue that the underdetermination claim is baseless, by ad-dressing each of these critiques. The first critique is addressed in Sec-tion 2 of the present article, whereas his pair of theorems are analyzedin detail in the sequel article [18].We note that the underdetermination attack is different from a pre-vious attack against hyperreal probabilities developed by AP in ([64],2014). In that paper, AP sought to argue that infinitesimals are“too small” to give plausible probabilities of individualoutcomes in a countably infinite lottery. [64, p. 1052]The 2014 attack was countered by Benci et al. ([12], 2018, Section 4.5,pp. 531–534). The more recent Prussian charge is unrelated to the For a discussion of the notions of internal and external entities see Section 3.1. See Sections 2.1 and 2.2 for technical details. Throughout the paper probability measures are assumed to be finitely additive .Notice that finitely additive probability measures include also σ -additive probabil-ity measures, but some infinite sample spaces admit finitely additive probabilitymeasures that are not σ -additive. Ironically, Reeder has criticized hyperreal infinitesimals for allegedly being too big in ([67], 2017). Reeder’s claim is refuted by Bottazzi ([15], 2019).
EMANUELE BOTTAZZI AND MIKHAIL G. KATZ “smallness” argument of [64]. Instead, AP argues that it is impossibleuniquely to assign an infinitesimal as the probability of an event. Asimilarity between [64] and [65] is that in both texts AP fails to take intoaccount the crucial distinction between internal and external sets andfunctions in Robinson’s framework. Meanwhile, the analysis in Benciet al. [12] is insufficient to address the underdetermination claim.AP asserts that such underdetermination is a feature of infinitesimalprobabilities generally, but his examples and theorems mainly focuson hyperreal-valued probability functions. Moreover, he suggests thatother non-Archimedean extensions of the real field could be more suit-able for the development of infinitesimal probabilities. Significantly, hemakes no attempt to present a model of his uniform processes in suchalternative frameworks with infinitesimals. We will address AP’s claimin Section 3 and in [18, Section 4].In Section 2.2 we point out some common hidden assumptions inmathematical modeling of physical phenomena, and analyze some com-mon biases against Robinson’s framework. Such biases include the fol-lowing: the insistence on the use of the natural numbers as the onlypossible model for the time scale of processes that “go on for the restof time” (Section 2.3) and the claim that uncountably many hypernat-ural numbers are not suitable for the representation of such discreteprocesses (Section 2.4).In Section 3 we highlight the significance of the transfer principleof Robinson’s framework. AP suggests that measures taking valuesin non-Archimedean fields other than hyperreal fields may be moresuitable for the development of infinitesimal probabilities. We showthat this suggestion overlooks the significance of the transfer principle.A common flaw of criticisms of infinitesimal models, as pursued byopponents of Robinson’s framework, is the assumption that certainproperties of the Archimedean accounts for infinite processes must alsobe satisfied by every non-Archimedean probability that represents it.However this assumption is unjustifiable; see [18, Section 2]. Moreover,in [18] we show that there are appropriate hyperfinite representationsof the process that are not underdetermined.The issues with the article by AP could be classified along the fol-lowing lines:(P1) (philosophical) (a) AP naively assumes that hyperreal mod-els must mimick the properties of Archimedean ones; see Sec-tions 2.3, 2.4, and [18, Section 2]. (b) AP fails to establish therelevance of the parasitic external measures he introduces tobuttress his underdetermination charge; see [18], Section 3.4.
NFINITE LOTTERIES, SPINNERS, APPLICABILITY OF HYPERREALS 5 (P2) (historical) AP ignores the Klein–Fraenkel criteria for the utilityof a theory of infinitesimals; see Section 3.3.(P3) (consistency) While at the outset AP admits that the arbitrari-ness claim against the hyperreals is mathematically incoherent,he lapses into it later in his article; see Section 3.4.(P4) (mathematical) AP makes inappropriate choices in hyperrealmodeling. Adopting more appropriate choices dissolves AP’sargument against hyperreal modeling; see [18, Section 2.4].In the present article we address items (P1), (P2), and (P3), whereasin [18] we address items (P1) and (P4).2.
On mathematical representation of physical processes
AP opens his analysis by articulating some “intuitions” based on sce-narios that can be represented by both Archimedean and non-Archime-dean probability measures (before going on to discuss his underdetermi-nation theorems that we will analyze in [18, Section 3]). Such scenarioscan be grouped into the following categories: • some infinite processes, such as coin tosses [65, Sections 3.2and 4.2], or the estimate of the value of a utility one can have“every day for eternity” [65, Section 3.5]; • a pair of uniform processes over a single sample space, exem-plified by the motion of two spinners (rotating pointers) [65,Sections 3.2 and 3.3] and by a pair of uniform lotteries over N [65, Section 4.1].We will discuss the details of AP’s representation of the two spin-ners in [18, Section 2]. Here we will comment more generally on theissue of mathematical representation of physical processes. We arguethat certain physical processes may admit distinct mathematical mod-els. For such processes, there does not exist a unique, well-definedmodel that would represent them in a way resembling anything like anisomorphism. Such a perspective is accepted even by mathematiciansand philosophers who adopt a responsible variety of mathematical re-alism, as discussed in Section 2.3 (see also Bottazzi et al. [17], 2019,Section 1). We further argue that hyperreal models can be used on par The example of an infinite collection of coin tosses has already been used to attackhyperreal-valued probability measures by Williamson [83], Easwaran [27], Parker[62] and other authors. Rebuttals of this argument can be found for instancein Weintraub ([82], 2008), Bascelli et al. ([10], 2014), Hofweber ([42] 2014), andHowson ([43], 2019).
EMANUELE BOTTAZZI AND MIKHAIL G. KATZ with Archimedean models based upon the Cantor–Dedekind represen-tations of the continuum. This issue is also dealt with by Herzberg([40], 2007, Section 4).As acknowledged by AP, a common objection to the use of hyperrealfields in such representations, namely that such fields are arbitrarilyspecified, can be countered in several ways. Such an objection can becountered either by working in the Kanovei–Shelah definable hyperrealfield [50] or by using suitable saturated hyperreal fields; see Section 2.1.In Sections 2.3 and 2.4 we address further hidden assumptions inArchimedean mathematical descriptions of the infinite processes rep-resented by coin tosses from the aforementioned point of view thatrejects the postulation of a unique mathematical representation. Inparticular, we argue that some commonly voiced objections to the useof hyperreals in the representation of scenarios involving an infinity ofevents stem from such hidden assumptions concerning mathematicalrepresentation of physical events.2.1.
Are hyperreal fields arbitrary?
AP acknowledges at the out-set the failure of the commonly voiced objection of arbitrariness (andeven that of ineffability ), namely that one cannot specify a particularhyperreal extension of R (see Section 2.2 for details). Such an objectionwas voiced by Alain Connes and others. The objection is specificallyrefuted by the Kanovei–Shelah definable hyperreal field [50]; see also[41]. For a rebuttal of the Bishop–Connes critique see Katz–Leichtnam([51], 2013), Kanovei et al. ([48], 2015), and Sanders ([72], 2020).Furthermore, AP acknowledges that the arbitrariness objection canalso be refuted by working with a hyperreal field defined up to anisomorphism, and that for suitable cardinals κ , there is a unique-up-to-isomorphism κ -saturated hyperreal field of cardinality κ . We willsource such “concessions” by AP in Section 2.2.2.2.
Pruss admits failure of arbitrariness/ineffability claims.
AP mentions a worry thatthe choice of a hyperreal extension appears to be notonly arbitrary but ineffable . . . – we cannot success-fully refer to a particular extension, and so a particularextension cannot reflect our credences . . . [65, Section 1] See ([22], 2004, p. 14) where Connes describes Robinson’s framework as“some sortof chimera.” More precisely, the condition is that an infinite cardinal κ should either be inac-cessible or satisfy 2 κ = κ + (that is, the continuum hypothesis holds at κ ). Fordetails see Keisler ([53], 1994, Section 11) and further references therein. NFINITE LOTTERIES, SPINNERS, APPLICABILITY OF HYPERREALS 7
However, he immediately acknowledges that “the ineffability argumentdoes not apply to all extensions of the reals, and even as restricted tothe hyperreals it is unsuccessful ” (ibid.; emphasis added).Thus AP acknowledges at the outset that the commonly voiced ob-jection of arbitrariness (and even that of ineffability ), namely the claimthat one cannot specify a particular hyperreal extension of R , is un-successful, and specifically refuted by the Kanovei–Shelah definablehyperreal field (see [50], [41]):[B]y leveraging the idea that even when it is difficultto specify a particular ultrafilter, one can specify setsof ultrafilters, Kanovei and Shelah (2004) explicitly de-fined a particular free ultrafilter on a particular infiniteset . . . Furthermore, Kanovei and Shelah used theirconstruction to make an explicitly specified extension ofthe reals (an iteration of the hyperreal extension usingthis ultrafilter) having further desirable properties. [65,Section 2]Moreover, AP acknowledges that the arbitrariness objection can also berefuted by working with a hyperreal field defined up to an isomorphism:[W]e can specify a set of hyperreals up to isomorphism.For some cardinals κ , there is a unique-up-to-isomor-phism κ -saturated non-standard real line of cardinal-ity κ . . . And there might be some non-arbitrary wayto choose the cardinal κ , perhaps a way matching theparticular problem under discussion. [65, Section 2]2.3. Shift-invariance hypothesis.
Various scenarios involving infi-nite processes have been discussed by AP and other authors, includingEaswaran, Parker, and Williamson. Such discussions often exhibit abias in favor of Archimedean models, which feature • a countable infinity of events, and • events that are ordered in time (rather than simultaneous).A typical process that is modeled with such hidden assumptionsis the outcome of an infinite amount of coin tosses. A number ofarguments against hyperreal probabilities for infinite coin tosses hingeupon the events H ( n ) that AP defines as follows: This characterisation by AP of the Kanovei–Shelah technique contains a mathe-matical inaccuracy. The technique does not exploit an ultrafilter on an infinite set.Rather, it exploits a maximal filter in a particular algebra of subsets of a certaininfinite set (the algebra in question does not contain all subsets!).
EMANUELE BOTTAZZI AND MIKHAIL G. KATZ starting with day n , it’s all heads for the rest of time.[65, Section 3.2]AP models such a process by a sequence of tosses labeled by N , andmakes the following assumption: Shift-invariance hypothesis : Events H ( n ) and H ( m )are isomorphic for all m, n ∈ N (the term shift-invariance hypothesis is ours). Meanwhile, alternativemodels of these infinite processes can be obtained with hyperfinite tech-niques, as discussed for instance by Benci et al. [11, pp. 44–46]; see alsoNelson ([61], 1987) and Albeverio et al. ([1], 1986). These models showthat the shift-invariance hypothesis is spurious, since it does not holdin a hyperfinite representation of the infinite collection of coin tosses. The shift-invariance hypothesis is often justified by an appeal to the“physical structure” of the infinite process. Thus, Williamson writes:“A fair coin will be tossed infinitely many times at one second intervals”in ([83], 2007) on page 174. By the middle of page 175, he is ready toclaim an ability tomap the constituent single-toss events of H (1 . . . ) one-one onto the constituent single-toss events of H (2 . . . )in a natural way that preserves the physical structure ofthe set-up just by mapping each toss to its successor.(ibid., p. 175; emphasis added)Williamson appears to be taking for granted a “physical structure”possessing a considerable supply of physical seconds.The same assumption appears in the more recent text by Parker([62], 2019). Parker implicitly assumes that a countable sequence ofcoin tosses is physically feasible, and bases his Isomorphism Principle [62, p. 4] on such an assumption.Bascelli et al. ([10], 2014) analyzed similar biases in favor of modelingbased upon a countable infinity of time-ordered events in Easwaran([27], 2014). Namely, an assumption of a countable time-ordered modelalready involves a full-fledged idealisation lacking a referent.What Williamson and Parker fail to recognize is that, even from theviewpoint of a responsible variety of mathematical realism, a mathe-matical description of a physical event typically involves some level ofidealisation and introduces some spurious properties (as already argued Howson argued that the events H ( n ) and H ( m ) are not equiprobable when-ever n = m even in the Archimedean model where the sample space is the σ -algebragenerated by the cylinder sets in { , } N ([43], 2019, Section 3). A similar observa-tion was made by Benci et al. [12, pp. 21–22]. Both arguments have been addressedby Parker [62]. NFINITE LOTTERIES, SPINNERS, APPLICABILITY OF HYPERREALS 9 for instance in [17, Section 1]). As a consequence of such idealisation,it is not possible to claim that a physical process has a unique well-defined sample space, or that one particular sample space provides theonly correct mathematical description of a physical process. For moredetails, we refer to [17, Section 1.4] and to references therein.In the coinflip case, it is obvious that it is physically impossible to flipa coin, as Parker would have it, “infinitely many times, at times t + n seconds for n = 0 , , , . . . ” [62, p. 8]. Nevertheless, it is possible to model this situation as a sequence of coin tosses over N , or with othernotions of number, as already mentioned.Furthermore, starting with a physical intuition of something going on“for the rest of time,” there are several possibilities of formalizing suchan intuition mathematically. One way is to interpret time incrementsas ranging over the traditional N . An alternative way of modeling suchan intuition would be to postulate that the “time” in question comesto an end rather than goes on indefinitely, given the likelihood of phys-ical armageddon expected by some modern theories in astrophysics. Ifso, then finite and hyperfinite modeling, which postulate such a finalmoment, are arguably more faithful to physical intuition than model-ing by N . In this sense, an assumption that an intuition of “for therest of time” necessarily refers to N , involves circular reasoning, as theconclusion is built into the premise. Attempting to base “intuitive rea-sons” against infinitesimal probabilities on such an idealized model, asAP does in [65, Section 3.2], begs the question as to why one assumesprecisely such a model rather than, say, a hyperfinite number of simul-taneous coin flips. Indeed, the model chosen by Easwaran, Parker,Pruss, and Williamson predetermines the outcome of their analyses.The shift-invariance hypothesis is analyzed further in Section 2.5.2.4.
Cardinality objection.
AP puts forth the following objectionto the use of hypernatural numbers:[I]t turns out that for any positive infinite number M in ∗ R , if ∗ R has a collection of hypernaturals, then therewill be uncountably many (in external cardinality) hy-pernaturals between 1 and M (Pruss 2014, Appendix). Or finite nonstandard number of flips in Nelson’s framework; see Section 3.2. AP’s parenthetical comment referring to external cardinality indicates that he isaware of the distinction between internal and external entities (in this case, car-dinality). Six years prior to the online publication of [65], he referred to internalcardinality in his posting [63]. However, AP tails to take into account the distinc-tion between internal and external hyperreal probabilities, as we will show in [18],Section 2.5.
And so the countable number of future days that we’veimagined is not what is counted by M : instead, M counts the number of members of the uncountable set { , , . . . , M } of hypernaturals. (Pruss [65], 2018, Sec-tion 3.5)What AP is claiming is that if one is interested in countably many“future days” (i.e., trials), hypernaturals do not provide an accuratemodel by cardinality considerations. However, his cardinality objectionis not valid, for the following reason. Skolem [80] already developedelementary (in the sense of PA) extensions N Sk of N in the 1930s.Skolem’s precedent was clearly acknowledged by Robinson ([69], 1966,pp. vii, 88, 278), who noted that “Skolem’s method foreshadows the ul-trapower construction” (op. cit., p. 88). Being built out of equivalenceclasses of (definable) sequences of natural numbers, N Sk naturally em-beds in ∗ R (for details see Kanovei et al. [49], 2013). If one’s interest isin countable structures only, one can proceed as follows:(1) construct a countable extension N Sk of N following Skolem;(2) form the field of fractions F of N Sk ;(3) F is then an ordered field properly including Q .In particular, there will be only countably many numbers in such an ex-tension F , and hence countably many numbers in the set { , , . . . , M } .An identical rebuttal applies to AP’s rejection of a nonstandard solu-tion to the paradox of Thomson’s lamp in ([66], 2018, p. 41).2.5. Standard model of the naturals and coinflipping.
In thissection we will examine the relation of the so-called standard modelof arithmetic to modeling infinite processes such as infinite lotteries,coin flips, etc. It is possible to disassociate the issue of scientific mod-eling (in physics, probability, etc.) from the issue of putative existenceof a standard model (a.k.a. the intended interpretation) of N . Evenmodulo such an N along the Cantor–Dedekind lines (not along theNelson lines), one can question the Pruss–Williamson (PW) assump-tion that N can be embedded in physical time. PW make no effort tojustify the assumption, which is surprising for publications in venuessuch as Analysis and
Synthese .In the following, we adopt the analysis of Kuhlemann ([55], 2018).When PW speak of performing a trial every second (or day) from nowto eternity and of the “physical structure” of the process, they maybe referring to either metalanguage natural numbers or the object lan-guage natural numbers.
NFINITE LOTTERIES, SPINNERS, APPLICABILITY OF HYPERREALS 11
Logicians make a distinction between, on the one hand, metalan-guage naturals, and, on the other, the object language naturals (e.g.,numbers in the putative standard model a.k.a. intended interpreta-tion). Thus, Simpson denotes metanaturals by ω to distinguish themfrom N [79, pp. 9–10]. At best, metanaturals can be related to as asorites-type subcollection (of the object language naturals N ) whichdoes not exist as a set, blocking implementation of a set of trials in-dexed by metanaturals (see [25, p. 255] for a related model of the soritesparadox).If PW mean to refer to metanaturals, the analysis above would un-dermine the shift-invariance hypothesis (see Section 2.3) and the PWattempt to identify H (1) with H (2) “naturally”. If, on the other hand, PW are referring to the object language nat-urals, then they are already making an assumption favoring one typeof idealisation over another, so that their conclusion is built into theirpremise. N is not naturally built into intuitions and thought experi-ments involving lotteries and coinflips (though it may be built into thetype of undergraduate mathematical training that PW received).3. On the strength of theories with infinitesimals
We will define the notions of internal and external objects in Sec-tion 3.1, and present the transfer principle of Robinson’s framework inSection 3.2. These notions play a major role in mathematical model-ing with hyperreal numbers. The significance of the transfer principleis also related to the historical development of mathematical theorieswith infinitesimals, as discussed in Section 3.3. Thus these notions willbe central to our discussion of the infinitesimal models of the uniformprocesses proposed by AP (see [18], Section 2), and of AP’s pair oftheorems (see [18], Section 3).We will also evaluate AP’s claim that transferless non-Archimedeanextensions of the real numbers might be more suitable for the devel-opment of infinitesimal probabilities. In Sections 3.3 and 3.4 we willelaborate on some consequences of the absence of transfer in the surrealnumbers and the Levi-Civita field. What this entails for their applica-bility or otherwise to probability theory is discussed in detail in [18],Section 4. Williamson actually uses the term natural in reference to applying the shift tophysical processes. AP actually speaks of “[t]he countable number of future days”[65, Section 3.5].
Constructing hyperreal fields.
It is well known that fields ∗ R of hyperreal numbers can be obtained by the so-called ultrapower con-struction . In this approach, one sets ∗ R = R N / U , where U is a nonprin-cipal ultrafilter over N . The operations and relations on ∗ R are definedfrom the quotient structure. For instance, given x = [ x n ] and y = [ y n ],we set x + y = [ x n + y n ] and x · y = [ x n · y n ]. We have x < y if and onlyif { n ∈ N : x n < y n } ∈ U .Let P = P ( R ) be the power set of R . Then the star transformproduces the object ∗ P . An internal subset A ⊆ ∗ R of ∗ R is by defini-tion a member of ∗ P . More concretely, in the ultrapower constructionan internal subset A ⊆ ∗ R is represented by a sequence ( A n ) of sub-sets A n ⊆ R . Here an element [ x n ] ∈ ∗ R belongs to A = [ A n ] if andonly if { n ∈ N : x n ∈ A n } ∈ U . A subset of ∗ R which is not internal iscalled external .More generally, in the context of the star transform from the super-structure over R to the superstructure over ∗ R , a set A of the latter isinternal if and only if it is a member of ∗ Z for some Z in the super-structure over R . For further properties of the ultrapower constructionof hyperreal numbers and the superstructures, see Fletcher et al. ([32],2017) and Goldblatt ([36], 1998).3.2. The transfer principle of Robinson’s framework.
Kanoveiet al. describe the transfer principle of Robinson’s framework asa type of theorem that, depending on the context, as-serts that rules, laws or procedures valid for a certainnumber system, still apply (i.e., are “transferred”) to anextended number system. ([47], 2018, p. 113)The transfer principle asserts that the internal objects of Robinson’sframework satisfy all the first-order properties of the correspondingclassical objects.The simplest examples of transfer involve the extension of sets andfunctions via the ∗ map. For instance, a continuous function f : R → R is a function that satisfies the formula ∀ x ∈ R ∀ ε ∈ R , ε > ∃ δ ∈ R , δ > ∀ x ∈ R ( | x − x | < δ → | f ( x ) − f ( x ) | < ε ) . The function f is extended to a function ∗ f : ∗ R → ∗ R that satisfies ∀ x ∈ ∗ R ∀ ε ∈ ∗ R , ε > ∃ δ ∈ ∗ R , δ > ∀ x ∈ ∗ R ( | x − x | < δ → | ∗ f ( x ) − ∗ f ( x ) | < ε ) NFINITE LOTTERIES, SPINNERS, APPLICABILITY OF HYPERREALS 13 (an internal function satisfying this formula is sometimes called ∗ conti-nuous). Moreover, ∗ f satisfies the Intermediate Value Theorem, theMean Value Theorem, and every other first-order property of f .We now turn to properties of sets under extension. For instance, theArchimedean property of R is expressed by the formula(3.1) ∀ x, y ∈ R (cid:0) (0 < x ∧ x < y ) → ∃ n ∈ N ( y < nx ) (cid:1) . An application of the transfer principle to the above formula yields(3.2) ∀ x, y ∈ ∗ R (cid:0) (0 < x ∧ x < y ) → ∃ n ∈ ∗ N ( y < nx ) (cid:1) . The latter formula needs to be distinguished from the former, since,as is well known, a ring extension of R with infinitesimal elements isnon-Archimedean. The first-order properties are preserved also by certain sets and func-tions that are not of the form ∗ X for some classical X . A relevantexample is given by a hyperfinite set , i.e., a set that can be put in an(internal) one-to-one correspondence with a set of hypernatural num-bers of the form { x ∈ ∗ N : x ≤ H } . Hyperfinite sets being internal,the transfer principle ensures that hyperfinite sets have the same first-order properties as finite sets. As a consequence, hyperfinite sets canbe routinely applied to a wide variety of problems. For a discussion ofselected applications, we refer to Arkeryd et al. ([2], 1997).Katz–Sherry ([52], 2013) suggest that the transfer principle can beinterpreted as a formalisation of the law of continuity of Leibniz; seealso Sherry–Katz ([78], 2014). Such a connection was first mentioned byRobinson ([69], 1966, p. 266). Robinson’s historical chapter 10 has oc-casioned a reappraisal of the legacy in infinitesimal analysis of pioneerslike Fermat [8], Gregory [9], Leibniz [3], Euler [4], and Cauchy [5], [6].We emphasize that the transfer principle applies only to internal entities of Robinson’s framework. Note that external entities do notexist in Nelson’s framework Internal Set Theory [60]. For this reason,attempted arguments from first principles that do not take into accountNelson’s framework are not actually based on first principles as they are The fundamental difference between the two formulas is that, in (3.2), the vari-able n can take infinite hypernatural values. In Nelson’s framework, set theory is enriched by a one-place predicate st . Theformula st ( x ) asserts that an entity x is standard. The ZFC axioms are enrichedby the addition of further axioms governing the interaction of the new predicatewith the axioms of traditional set theory. Infinitesimals, say ǫ , are found withinthe ordinary real line, and satisfy 0 < | ǫ | < r for all standard x ∈ R + . It is shownin [60] that the new theory is conservative with respect to ZFC. A related systemwas developed independently by Hrbacek [44]. For further details, see Fletcher etal. ([32], 2017) and Hrbacek–Katz ([45], 2020). claimed to be, but rather involve an unspoken commitment to a specificset-theoretic framework (for instance, Zermelo–Fraenkel set theory plusthe Axiom of Choice) expressed in the ∈ -language. This is done at theexpense of other possible foundational frameworks. Thus, from thepoint of view of Internal Set Theory, internal probability measures areno less underdetermined than the traditional ones.3.3.
Usefulness of infinitesimals: Klein and Fraenkel.
Duringthe opening decades of the 20th century, both Felix Klein ([54], 1908)and Abraham Fraenkel ([35], 1928) formulated a pair of criteria togauge the success of theories with infinitesimals. These criteria are(1) the availability and provability (by infinitesimal techniques) ofthe Mean Value Theorem, and(2) the introduction of the definite integral in terms of infinitesimalincrements.For a detailed discussion, see Kanovei et al. ([47], 2018). Klein andFraenkel both observed that the infinitesimal theories available at thetime (including the Levi-Civita field) did not enable a satisfactory treat-ment of these topics. When Robinson introduced his framework foranalysis with infinitesimals, Fraenkel related to Robinson’s frameworkas an important accomplishment that finally solved the old problemof developing a usable non-Archimedean field. Thanks to the trans-fer principle, in Robinson’s framework it is possible to prove the MeanValue Theorem (MVT) and to define the Riemann integral of a continu-ous function by means of hyperfinite summation of infinitesimal terms.Notice that the MVT and the definition of an integral, and in gen-eral the development of a calculus on non-Archimedean structures thatextends the real calculus, require an extension of real functions. Costinet al. observe thatA longstanding aim has been to develop analysis on [thesurreal numbers] as a powerful extension of ordinaryanalysis on the reals. This entails finding a natural wayof extending important functions f from the reals to thereals to functions f ∗ from the surreals to the surreals,and naturally defining integration on the f ∗ . [23, Ab-stract] More specifically, the unspoken commitment typically involved is to a set theoryin the ∈ -language rather than a set theory in the ∈ - st -language; see note 16. In recent decades, there has been a deplorable attempt by Mehrtens ([58], 1990),Gray ([38], 2008), and others to discredit Klein both mathematically and politically.A rebuttal appears in Bair et al. ([7], 2017).
NFINITE LOTTERIES, SPINNERS, APPLICABILITY OF HYPERREALS 15
In Robinson’s framework, such an extension is provided by the ∗ map.Meanwhile, it is still an open problem to define well-behaved extensionsof real functions to the surreals or to the Levi-Civita field. For the sur-real numbers, the problem is caused by the necessity to define functionsfrom the simplicity hierarchical structure of the surreal number tree. Meanwhile, every real continuous function admits a canonical exten-sion to a continuous function on the Levi-Civita field. However, thisextension does not preserve many properties of the original real con-tinuous functions, such as an Intermediate Value Theorem or a MeanValue Theorem. Moreover, the surreal field and the Levi-Civita field satisfy only somerestricted versions of the Klein–Fraenkel criteria. Namely, in the sur-real numbers it is not possible to prove the MVT, and it is still an openproblem to define an integral (see for instance Costin et al. [23] and For-nasiero [34]). Meanwhile, for the Levi-Civita field, Shamseddine showedthat the MVT is valid only for analytic functions [73]. Consequently,this theorem fails for instance for the extension of non-analytic realcontinuous functions. The Levi-Civita field does have a notion of inte-gral in dimensions 1, 2 and 3, but the integral is not defined in terms ofsums of infinitesimal contributions; see Berz–Shamseddine ([13], 2003),Shamseddine ([75], 2012), Shamseddine–Flynn ([77], 2018). In addi-tion, it turns out that the extensions of continuous but non-analyticreal functions are not measurable (Bottazzi [16]).When Klein and Fraenkel formulated their criteria for the evalua-tion of non-Archimedean extensions of the reals (see [47]), a numberof such non-Archimedean options were available, including the Levi-Civita field. It is those ordered fields that Klein and Fraenkel werereferring to when they expressed disappointment with the (then) cur-rent rate of progress, when even the Mean Value Theorem was notyet provable using infinitesimal analysis. AP’s critique of Robinson’sframework fails to take into account the fact that at present, Robin-son’s is the only theory of infinitesimals that meets the Klein–Fraenkel For instance, no surreal extension of the sine function usable in ordinary math-ematics is as yet available; see e.g., Kanovei’s remark at https://mathoverflow.net/a/307114 In this context, Bottazzi suggested an analogy between these extensions and ex-ternal functions of Robinson’s framework [14]. Recall that every real continuous function is Lebesgue-measurable and, if it isdefined over a closed interval, it also has a well-defined Riemann integral. The fail-ure of measurability for the extensions of non-analytic real functions is a significantlimitation for the measure theory on the Levi-Civita field and it is also a blatantfailure of transfer for this field. criteria of utility. These criteria are prerequisites for a measure orprobability theory.3.4.
Infinitesimals without transfer.
The importance of the trans-fer principle in non-Archimedean extensions of the real numbers canbe better appreciated if one considers what happens when this princi-ple is not available. We will refer to such non-Archimedean fields as transferless .Thus, in Henle’s non-nonstandard analysis [39] what is available isa weak form of transfer that applies only to equations, inequalities andtheir conjunctions, but not to their disjunction. As a consequence,some properties of ordered fields fail, and Henle’s extension is only apartially ordered ring with zero divisors. Similarly, in the Levi-Civita field the absence of a transfer principlemakes it necessary to prove individually many theorems of the cal-culus, such as the Intermediate Value Theorem and the Mean ValueTheorem for analytic functions. For a more detailed discussion, seeShamseddine–Berz ([76], 2010) and Shamseddine ([73], 2011). In ad-dition, currently it is possible to extend only analytic real functions tothe Levi-Civita field in a way that preserves their first-order properties,as shown by Bottazzi ([14], 2018).In other transferless fields the situation might be even more difficult;we are not aware of any research towards establishing some (even lim-ited) forms of transfer in such settings. Nevertheless, AP suggests thatthere are “multiple methods” of developing infinitesimal probabilitiesin transferless fields:[H]yperreals are not the only way to get infinitesimals.There are multiple methods that do not make use ofanything like the arbitrary choice of an ultrafilter. [65,Section 2]AP argues that, as a consequence,The friend of infinitesimal probabilities has a real hopeof non-arbitrarily specifying a particular field of infinites-imals. (ibid.) In some cases, working with number systems lacking the habitual properties canlead the author into error. Thus, Laugwitz pointed out that Henle’s article lapsesinto using denominators when working with a ring. Laugwitz goes on to invite thereader to “rewrite the relevant passages” [56]. Here AP seems to lapse into the arbitrariness charge against Robinson’s frame-work discussed in Section 2.1. Notice that this passage comes after AP’s admissionthat it is possible to uniquely specify some particular hyperreal fields.
NFINITE LOTTERIES, SPINNERS, APPLICABILITY OF HYPERREALS 17
Moreover, he claims that certain transferless fields, namely the surrealnumbers, fields of Laurent series or the Levi-Civita field, have someadvantages over hyperreal fields. Thus AP writes:[T]he surreals have the advantage of being exhaustivelylarge, large enough that they escape the cardinality ar-guments against regularity of Pruss (2013a). The fieldsof formal Laurent series and the Levi-Civita field, onthe other hand, have the advantage of being elegantlysmall. (On the other hand, the Kanovei–Shelah field,while mathematically fascinating, probably has little go-ing for it in this context.) [65, Section 2]It can be argued that the main advantage of the field of Laurent seriesand the Levi-Civita field is that of being definable from the naturalnumbers in a choice-free manner. If one works in the von Neumann–Bernays–G¨odel set theory with global choice, then the surreal numbersare also uniquely specified. It is also true that many hyperreal fieldsobtained as ultrapowers of R are not definable in a choice-free manner;however, as AP acknowledges, suitable κ -saturated fields of hyperrealnumbers are uniquely specified up to an isomorphism. However, the issue at hand is not whether a non-Archimedean fieldis definable from the natural numbers without additional parameters.The real issue is the applicability of such a field. In this regard, theLevi-Civita field has some limited applications in automatic derivation[76] and in the description of physical phenomena [33], while there is avast literature of applications of hyperreal fields. That such applications are possible is due in particular to the transferprinciple of Robinson’s framework. Significantly, the principle is notdiscussed by AP in [65]. Instead, AP expresses enthusiasm aboutthe surreal numbers, Laurent series, and Levi-Civita fields, but fails toexplain their shortcoming, namely lack of transfer. Significantly, APdoes not develop a model either for the infinite coin tosses or for hisspinners in any of these transferless structures. Thus, his claim that AP’s parenthetical claim that the KS model has little advantage over the trans-ferless fields mentioned is questionable, since the KS model can be used just as wellas any traditional R N / U model (see Section 3.1), particularly with the saturationimprovement provided by Kanovei–Shelah [50]. In particular, the KS model hasthe advantage of the transfer principle. See Section 2.1. Some relevant examples can be found in note 1. It is a matter of public record that AP is aware of the transfer principle, since hementions it both in [65, Appendix] and in [66, p. 41]. these non-Archimedean fields may be suitable for the development ofan infinitesimal probability is baseless.Indeed, in [18, Section 4] we argue that attempts to develop infini-tesimal probabilities over the surreal numbers or over the Levi-Civitafield encounter a number of difficulties.For instance, in order to accomplish anything with the surreals onewould have first to import the transfer principle via an identification ofmaximal class-size surreals with maximal class-size hyperreals, as men-tioned by Ehrlich [30, Theorem 20]. Without such an identification andwithout the transfer principle of Robinson’s framework, it is currentlynot possible to develop a measure theory on the surreal numbers (formore details, see [18, Section 4.1]). The Levi-Civita field does obey a type of transfer principle, albeitlimited to a particular extension of real analytic functions; see Bottazzi([14], 2018).One could also work directly with the measure-theoretic tools avail-able in the various theories to define non-Archimedean probability mea-sures; however, there seem to be some difficulties.Consider for instance the case of the Levi-Civita field, where a uni-form measure is currently under development by Shamseddine and Berz[13], Shamseddine [75], Shamseddine and Flynn [77] and Bottazzi [16].So far this uniform measure is not able to accommodate more thanlocally analytic probability functions, and has no notion of hyperfinite-ness comparable to that of Robinson’s framework. This example isdiscussed further in [18], Section 4.2.4.
Conclusion
We have examined some commonly voiced objections to the use of hy-perreal numbers in mathematical modeling. Many of these objectionsare based upon naive assumptions regarding the possibility of uniquelyspecifying some hyperreal fields, and upon examples of infinite pro-cesses that allegedly do not allow for a uniquely defined infinitesimalprobability for singletons.The first objection, namely that it is allegedly not possible to spec-ify a hyperreal field in a unique way, is refuted by the Kanovei–Shelahdefinable hyperreal field and by the fact that, for suitable infinite car-dinals κ , there is a unique-up-to-isomorphism κ -saturated hyperreal It should be noted that the omnific surnaturals do not satisfy the axioms of PeanoArithmetic; e.g., there exist surnaturals p, q such that p = 2 q . For further detailssee [59], [71], [46]. NFINITE LOTTERIES, SPINNERS, APPLICABILITY OF HYPERREALS 19 field of cardinality κ . Note that this rebuttal is accepted also by somedetractors of Robinson’s framework for analysis with infinitesimals.With regard to objections based upon the analysis of certain infi-nite processes, we have observed that physical processes often admitdistinct mathematical models, so that there does not exist a unique,well-defined model that would represent them in a way resemblinganything like an isomorphism. Thus we have shown that, for somecommonly used models e.g., of infinite coin tosses, some objections tothe use of Robinson’s framework stem from hidden and unnecessaryassumptions that predetermine the choice of an Archimedean model.Dropping such hidden assumptions enables alternative models of theseinfinite processes, obtained via hyperfinite techniques.Moreover, we have started addressing the claim by Pruss that trans-ferless extensions of the real numbers (such as the surreal field, the Levi-Civita field, or the field of Laurent series) might be more suitable forthe development of infinitesimal probabilities. The proposal of work-ing with such non-Archimedean fields ignores both the Klein–Fraenkelcriteria for gauging the applicability of theories with infinitesimals, andthe power and utility of the transfer principle. In [18] we show that,due to these limitations, the measure theory on such transferless fieldsis less expressive than the hyperfinite counting measures.Pruss claims that “[w]hatever you can do with hyperreals, you cando with surreals” and that “[t]he fields of formal Laurent series and theLevi–Civita field . . . have the advantage of being elegantly small” [65].However, in [18] we will see that, by his own Theorem 1, these fields suf-fer from underdetermination due to the possibility of rescaling the infin-itesimal part of the probability, and do not possess a notion of internal-ity that enables one to escape such underdetermination in Robinson’sframework. Prussian Theorem 1, when properly analyzed, boomerangsto undercut his own underdetermination thesis. Prussian Theorem 2is similarly Boomerang 2 due to the existence of nontrivial automor-phisms for all such transferless fields. Arguments based on the non-effectiveness of ultrafilters are not lim-ited to the work of Pruss; see e.g., Easwaran–Towsner [28]. In spite ofan initial plausibility of such arguments against Robinson’s framework,the arguments dissolve upon closer inspection, and even tend to provethe opposite of what their authors intended. One can well wonder See Ehrlich ([29], 1994, pp. 253) for the existence of nontrivial automorphisms ofthe surreals as an ordered field, and Shamseddine ([74], 2011, p. 224, Remark 3.17)for such existence for all ordered extensions of R . why such arguments ad ultrafiltrum don’t succeed. A recent develop-ment suggests a possible reason. It turns out that the main body ofthe applicable part of Robinson’s framework admits a formalisationthat requires modest foundational means not exceeding those requiredfor traditional non-infinitesimal methods in ordinary mathematics; seeHrbacek–Katz [45]. Thus the alleged non-effectiveness is simply notthere to begin with. Acknowledgments
We are grateful to Karel Hrbacek, Vladimir Kanovei, Karl Kuhle-mann, and David Sherry for insightful comments on earlier versionsthat helped improve our article, and to anonymous referees for con-structive criticism. The influence of Hilton Kramer (1928–2012) is ob-vious.
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E-mail address : [email protected], [email protected] M. Katz, Department of Mathematics, Bar Ilan University, RamatGan 5290002 Israel, orcid 0000-0002-3489-0158
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