Internality, transfer, and infinitesimal modeling of infinite processes
aa r X i v : . [ m a t h . HO ] A ug INTERNALITY, TRANSFER, AND INFINITESIMALMODELING OF INFINITE PROCESSES
EMANUELE BOTTAZZI AND MIKHAIL G. KATZ
Abstract.
A probability model is underdetermined when thereis no rational reason to assign a particular infinitesimal value asthe probability of single events. Pruss claims that hyperreal prob-abilities are underdetermined. The claim is based upon exter-nal hyperreal-valued measures. We show that internal hyperfinitemeasures are not underdetermined. The importance of internalitystems from the fact that Robinson’s transfer principle only appliesto internal entities. We also evaluate the claim that transferlessordered fields (surreals, Levi-Civita field, Laurent series) may haveadvantages over hyperreals in probabilistic modeling. We showthat probabilities developed over such fields are less expressive thanhyperreal probabilities.
Contents
1. Introduction 22. Archimedean and otherwise models of Pruss spinners 32.1. Discrete models of the spinners 52.2. Spinner models based on the Lebesgue measure 62.3. Hyperfinite models of the spinners 72.4. Hidden assumptions on infinitesimal models of spinners 92.5. Refining the models by means of additional constraints 93. Are infinitesimal probabilities underdetermined? 123.1. The first theorem of Pruss 133.2. The Ω-limit axiom 143.3. The second theorem of Pruss 153.4. Internal probability measures are not underdetermined 164. Probability measures on transferless fields 174.1. Probability measures on the surreal numbers 17
Date : August 27, 2020.2010
Mathematics Subject Classification.
Primary 03H05; Secondary 03H10,00A30, 26E30, 26E35, 28E05, 60A05, 01A65.
Key words and phrases.
Infinitesimals; hyperreals; hyperfinite measures; internalentities; probability; regularity; axiom of choice; saturated models; underdetermi-nation; non-Archimedean fields.
Introduction
Alexander Pruss (AP) claims in [28] that hyperreal probabilities areunderdetermined, meaning that, given a model, there is no rationalreason to assign a particular infinitesimal value as the probability of asingle event. AP’s underdetermination claim hinges upon the following:(1) examples of uniform processes that allegedly do not allow for auniquely defined infinitesimal probability for singletons, and(2) a pair of theorems asserting that for every hyperreal-valuedprobability measure there exist uncountably many others thatinduce the same decision-theoretic preferences.In [12] we presented our main arguments highlighting some hiddenbiases in the representation of some infinite processes such as lotteries,coin tosses, and other infinite processes.In Section 2 we analyze some Archimedean and non-Archimedeanmodels of Prussian spinners (rotating pointers). In particular, we willshow that hyperfinite models of the spinners are not underdetermined.Moreover, we will discuss additional constraints that may narrow downthe choice of infinitesimal probabilities, as proposed by AP himself in[28], Section 3.3 as well as 3.5 (“Other putative constraints”).We show that the additional probabilities introduced in AP’s theo-rems are all external; see Section 3. Since external functions do notobey the transfer principle of Robinson’s framework, these additionalmeasures are inferior to internal ones when it comes to modeling usinghyperreal fields. Thus AP’s external measures are parasitic in the senseof Clendinnen ([14], 1989), and fail to establish underdetermination.In the light of our analysis, AP’s theorems can be framed as a warningagainst the use of external probabilities that do not satisfy the transferprinciple. Thus, among the additional constraints that, in AP’s words,“may help narrow down the choice of infinitesimal probabilities” ([28,Section 3.3]), the first choice should be the internal constraint; seeSections 2.5 and 3.4 as well as [12], Section 3.2. AP speculates that See [12], note 2.
NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 3 [An] approach in terms of imprecise probabilities, wherea family of hyperreal-valued probability functions is as-signed as in Sect. 6.1.2 of Benci et al. (2018), may wellbe able to escape the underdetermination worries. ([28,Section 1]; emphasis in the original)AP’s wording implies that a single hyperreal-valued probability is un-able to escape such ‘worries.’ However, we show that a single, internalhyperreal-valued probability does escape Prussian worries.AP also speculated that other non-Archimedean fields may bestowadvantages as compared to Robinson’s framework. In Section 4 weargue that, contrary to AP’s speculation, infinitesimal probabilities inother familiar non-Archimedean extensions of the real numbers are lessexpressive than real-valued probabilities. This is due to the absence ofa uniform way of extending most real functions to these fields so as topreserve their elementary properties, and to the limitations of the mea-sure theory currently developed for these fields. From the viewpoint ofRobinson’s framework, such limitations are due to the unavailability ofa transfer principle for these fields.2.
Archimedean and otherwise models of Pruss spinners
We analyze AP’s non-Archimedean models of the spinners, and showhow hyperfinite non-Archimedean models avoid the claimed underde-termination problem. Starting from some of these hyperfinite models,it is also possible to define an infinitesimal probability with a stan-dard sample space and taking values in a field of hyperreals ∗ R , thusaddressing AP’s original model (see Section 2.5). In this section we ex-amine one particular example of the failure of AP’s underdeterminationcharge for hyperreal probabilities. Our rebuttal of his claim that every non-Archimedean probability is underdetermined appears in Section 3.AP provides the following example (see [28, Section 3]) of a pair ofuniform processes with a common sample space:(1) the first process is a spinner that is designed to stop uniformlyat an angle θ ∈ [0 , θ ∈ [0 , , EMANUELE BOTTAZZI AND MIKHAIL G. KATZ we cannot simply specify an infinitesimal [probability]by saying that it is whatever is the probability of a uni-form process hitting a particular point. For what thatprobability is – assuming it is infinitesimal and not zero– depends on details of the process that go beyond uni-formity. [28, Section 3.2]To refute AP’s claim, we will show that his conclusion is only dueto inappropriate choices he makes in the non-Archimedean descriptionof the uniform process, and that indeed appropriate non-Archimedeanmodels do not depend upon irrelevant details.We observe that AP appears to extrapolate to every non-Archime-dean model some properties of the continuous model obtained from theLebesgue measure. For instance, he assumes that in every infinitesimaldescription of the spinners, the sample space S must have the propertythat if x ∈ S , then also x ∈ S . This property is satisfied if S is aninterval of Q or of R , but the spinners can also be described by meansof a hyperfinite set S H that is discrete in an appropriate sense. Therepresentations of the uniform processes by means of a hyperfinite sam-ple space S H enable us to specify a unique uniform probability measureover S H that is also regular (see also Section 2.3). Thus for such hyper-real probabilities no underdetermination occurs. In Sections 3.2 and 3.3we argue that this result applies also to non-Archimedean probabilitiesthat satisfy the Ω-limit axioms of Benci et al., since these probabilitiescan be obtained as the restriction of suitable hyperfinite measures.Consequently, in his intuitive arguments for his underdeterminationclaim AP relies on additional hypotheses on the non-Archimedean mod-els that are not satisfied by many relevant hyperreal probabilities.In Archimedean mathematics, the two spinners discussed by AP canbe described by a family of discrete models and by a continuous model,as we explain in Sections 2.1 and 2.2. In Sections 2.3 and 2.4 weshow how internal hyperfinite models of AP’s spinners avoid the al-leged drawbacks. Finally, in Section 2.5, we show how properly definedhyperfinite measures can approximate to varying degrees other proper-ties of the models based upon the Cantor–Dedekind representation ofthe continuum. We have already argued that a given physical process can be given distinct mathe-matical representations; see [12], Section 2.3. Thus we reject the claim that chang-ing the sample space, as we have proposed here, changes also the underlying process.
NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 5
Discrete models of the spinners.
The discrete models for thefirst spinner can be obtained as follows. Let n ∈ N and define S n = (cid:8) , . . . , kπn , . . . , (2 n − πn (cid:9) ⊆ [0 , π ) . One can imagine the set S n as representing the points on the circleobtained from rotations by integer multiples of the angle πn . A uniformprobability over S n is given by P n ( A ) = | A || S n | = | A | n . The probabilitymeasure P n has the following four properties: Tot P n is total, in the sense that it is defined on the powerset P ( S n )of the sample space S n ; Un n P n is uniform in the sense that for every A, B ⊆ S n with | A | = | B | , one has also P n ( A ) = P n ( B ); Reg P n is regular; ¬ Sy R P n is not symmetric with respect to rotations by an arbitraryreal angle, since if θ ∈ R in general kπn + θ mod 2 π S n .In addition, as a consequence of property Un n , the measure P n satisfiesthe following discrete symmetry condition: Sy n P n symmetric with respect to rotations by multiples of πn , i.e.if 0 ≤ θ ≤ n − P (cid:0)(cid:8) k πn , . . . , k j πn (cid:9)(cid:1) = P (cid:16)n k πn + θ πn mod 2 π, . . . , k j πn + θ πn mod 2 π o(cid:17) . The discrete models Q n for the second spinner can be obtained fromthe measure P n as follows: Q n ( { x } ) = P n ( { x } ) + P n ( { x + π } ) = 2 n = n = P n ( { x } )for all x ∈ S n . From the previous equation it is readily seen that Q n agrees with P n over S n , even though it is obtained from another descrip-tion of the uniform process. As a consequence, Q n has the properties Un n , Reg , ¬ Sy R and Sy n already discussed for P n .As already observed by Bascelli et al. ([2], 2014), with a choice ofsome sufficiently large n , the models P n and Q n might already be suf-ficient for many practical purposes, sinceall physical quantities can be entirely parametrized bythe usual rational numbers alone, due to the intrinsiclimits of our capability to measure physical quantities.[2, p. 853] By regular we mean that the measure P n assigns probability 0 only to the emptyevent. This should not be confused with the notion of regular measure , i.e. of ameasure ν such that ν ( A ) = sup { ν ( F ) : F ⊆ A is compact and measurable } =inf { ν ( G ) : G ⊇ A is open and measurable } . EMANUELE BOTTAZZI AND MIKHAIL G. KATZ
This position is shared by Herzberg [21, Section 1]. Nevertheless, weagree with the common mathematical practice that it is more conve-nient to introduce some level of idealisation for the description of P n and of Q n , especially as n becomes very large.2.2. Spinner models based on the Lebesgue measure.
A typicalidealisation is the use of continuous models based upon the real num-bers. A continuous model for the first spinner possessing the followingproperty, Un , of uniformity: Un if two intervals have the same length, then they have the sameprobabilityis obtained by using the Lebesgue measure µ over the real interval[0 , π ). In this model, events are the measurable subsets of [0 , π ) andthe probability of an event A is defined as P µ ( A ) = µ ( A ) µ ([0 , π )) = µ ( A )2 π .Notice also that the choice of [0 , π ) as the sample space is arbi-trary. One could equally well measure angles with arc degrees in-stead of radians. In this case, the sample space would be the realinterval [0 , A would be definedas P ( A ) = µ ( A ) µ ([0 , = µ ( A )360 . Despite the arbitrariness of the choice ofthe sample space, one obtains the compatibility conditions(2.1) P ( A ) = P µ (cid:0)(cid:8) x ∈ [0 , π ) : π x ∈ A (cid:9)(cid:1) for all measurable sets A ⊆ [0 , P µ ( B ) = P (cid:0)(cid:8) x ∈ [0 , π x ∈ B (cid:9)(cid:1) for all measurable sets B ⊆ [0 , π ). From now on, we will refer mostlyto the probability measure P µ , but the discussion remains valid alsofor P . The probability measure P µ has different properties from itsdiscrete counterparts P n . Namely, it has the following three properties: Un P µ is uniform; ¬ Reg P µ is not regular; Sy R P µ is symmetric with respect to rotations by an arbitrary realangle. Observe that we have not taken a position on whether the continuous modelssatisfy
Tot or ¬ Tot : if one rejects the full Axiom of Choice, then the Lebesguemeasure can be total; see Solovay ([35], 1970). However, by assuming a suffi-ciently strong choice principle it is possible to prove that there are sets that are notLebesgue measurable, so that P µ would not be total. Recall also that without someweak form of the Axiom of Choice it is not possible to prove that the Lebesguemeasure is countably additive. For more details, we refer to Fremlin ([20], 2008)and to Bottazzi et al. ([11], 2019, Section 5). NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 7
Notice also that for all n ∈ N , and for all h < k ≤ n , we have P µ (cid:0)(cid:2) hπn , kπn (cid:1)(cid:1) = P n (cid:16)n hπn , . . . , ( k − πn o(cid:17) . The previous equality can be interpreted as a coherence between P µ and P n over arcs with endpoints in S n .Similarly, the second spinner can be modeled by using the Lebesguemeasure. The probability of an event A is Q µ ( A ) = µ ( { x : 2 x mod 2 π ∈ A } ) µ ([0 , π ))= µ ( { x/ x ∈ A } )2 π + µ ( { x/ π : x ∈ A } )2 π . Since the Lebesgue measure is translation invariant, we have the equal-ity µ ( { x/ x ∈ A } )2 π = µ ( { x/ π : x ∈ A } )2 π , from which we deduce Q µ ( A ) = 2 µ ( { x/ x ∈ A } )2 π = 2 P µ ( { x/ x ∈ A } ) = P µ ( A ) . The discrete models and the continuous model capture different as-pects of the spinners. The choice of which model to use in specificcircumstances ought to reflect the relevant properties of each situation.For instance, one should take into account that, as is well known, theproperties Un , Reg and Sy R cannot be simultaneously satisfied byany real-valued measure defined over an infinite set Ω. This imposessome limitations on the scope of P . For instance, it is not possible todefine from P a conditional probability with respect to every measur-able event B . This occurs since the common definition of conditionalprobability P ( A | B ) = P ( A ∩ B ) P ( B ) requires B to have positive measure.2.3. Hyperfinite models of the spinners.
The continuous modelbased upon the Lebesgue measure is not the only possible idealisationof the two discrete models for the spinners. By using hyperfinite rep-resentations, it is possible to define models that satisfy
Tot and
Reg ,and that approximate to varying degrees the properties Un and Sy R .A simple model of the first spinner is obtained by taking as sam-ple space the set S n for some infinitely large hypernatural n . Thecorresponding probability measure P n has the properties Tot , Un n , Such models exist thanks to the transfer principle of Robinson’s framework. More-over, transfer ensures that the properties of the finite models discussed in Section 2.1are shared by the hyperfinite models presented here.
EMANUELE BOTTAZZI AND MIKHAIL G. KATZ
Reg , ¬ Sy R , Sy n described for the discrete models of the spinner. If n = m ! for some infinite hypernatural m , then Sy n implies Sy Q ,that is symmetry with respect to rotations by any rational angle. Asobserved for instance by Benci et al. [3, p. 5], P n cannot satisfy theproperties Un and Sy R ; however it has the coherence property Co P n ( ∗ A ∩ S n ) ≈ P µ ( A ) for each measurable A ⊆ [0 , π ), that implies the weak form of uniformity P n ([ a, b )) ≈ P n ([ c, d )) whenever b − a = d − c, and the weak form of symmetry P n ( ∗ A ∩ S n ) ≈ P n ( ∗ { x + θ mod 2 π : x ∈ A } ∩ S n )for every measurable A ⊆ [0 , π ) and for every real θ ∈ [0 , π ).As already discussed in the Archimedean discrete models, the secondspinner can be represented by the hyperfinite probability Q n for aninfinite hypernatural n . This probability measure shares the properties Tot , Un n , ¬ Sy R , Sy n and Co with the first spinner; moreover onehas Q n ( { x } ) = P n ( { x } ), as in the Archimedean models.Recall that the continuous description of the spinners based uponthe real Lebesgue measure could be formulated with different choicesof the sample space (e.g., the interval [0 , π ) or the interval [0 , P n we have a similarproperty, expressed by the following compatibility condition: when-ever gcd( m, n ) = min { m, n } = n , we have(2.3) P n ( A ) = P m (cid:0)(cid:8) x ∈ S m : mn x mod 2 π ∈ A (cid:9)(cid:1) for all A ⊆ S n .Equality (2.3) is the discrete counterpart of equations (2.1) and(2.2). The main differences between the continuous equations andthe hyperfinite one is that in the real case there is a bijection be-tween the sets [0 , π ) and [0 , x π x , whilein the discrete case (either finite or hyperfinite) there is no bijection Property
Tot should be interpreted as follows: P n is defined on every internal set A ⊆ S n . Similarly, property Un n should be interpreted as follows: for every internal A, B ⊆ S n with | A | = | B | , one has also P n ( A ) = P n ( B ). The symbol | A | denotes the element of ∗ N corresponding to the internal cardinality of the internalset A . Here ≈ denotes the relation of infinite proximity, i.e., the relation of being infinitelyclose. NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 9 between the sets S n and S m unless n = m . On the other hand,if gcd( m, n ) = min { m, n } = n , then there is a mn -to-one correspon-dence between S m and S n ; together with properties Un n and Un m ofthe measures P n and P m , this is sufficient to entail the relation (2.3).2.4. Hidden assumptions on infinitesimal models of spinners.
In his Section 3.2, AP seems to suggest that in Robinson’s frameworkevery pair of probability measures P and Q for the first and the secondspinner, respectively, satisfy the following conditions:(Pr 1) P and Q have the same sample space;(Pr 2) P and Q should assign the same probability to singletons;(Pr 3) Q ( { x } ) = 2 P ( { x/ } ).From properties (Pr 2) and (Pr 3) AP obtains his underdetermination claim [28, Section 3.2].However, as we have seen in the discussion of the discrete models,if one wishes to retain property (Pr 1) in a hyperfinite setting, the re-quirement (Pr 3) must be replaced by a subtler condition. In particular,for the hyperfinite probabilities P n and Q n defined on S n we have seenthat Q n should be defined in terms of P n as follows:(3 ′ ) Q n ( { x } ) = P n ( { x } ) + P n ( { x + π mod 2 π } ) = 2 P n ( { x } ) , since for some x ∈ S n the number x/ S n but does belong to S n . We have already argued thatdefinition (3 ′ ) implies that Q n ( { x } ) is indeed equal to P n ( { x } ) for ev-ery x ∈ S n . This example shows how internal hyperfinite models of thetwo spinners are sufficient to refute AP’s claim that the infinitesimalprobability of a pair of uniform processes on the same outcome spaceallegedly “depends on details of the process that go beyond uniformity”[28, Section 3.2].A similar rebuttal applies to AP’s description of the pair of uniformlotteries over N in [28, Section 4.1].2.5. Refining the models by means of additional constraints.
AP notes that We remark that there is no internal bijection between S n and S m . If one drops therequirement that the bijection must be internal, then it is possible to find externalbijections witnessing that the external cardinality of S n is equal to the externalcardinality of S m whenever both n and m are infinite. However, the existence ofexternal bijections between S n and S m has no bearing our argument, that is insteadbased upon the internal cardinality. For a discussion on the limited relevance ofexternal objects and functions in hyperreal models see Section 3 and note 22, aswell as [12], Sections 3.2 and 3.4. [T]here are potential types of constraints on the choiceof infinitesimals that may help narrow down the choiceof infinitesimal probabilities. One obvious constraint isformal: the axioms of finitely additive probability. Asecond type of constraint is match between the extendedreal probabilities for a problem and the correspondingclassical real-valued probabilities. ([28, Section 3.3]; em-phasis added)Thus, AP envisions the possibility of introducing additional constraintsthat may narrow down the choice of infinitesimal probabilities. He evenevokes more specifically the possibility of exploring a “match” betweenextended-real probabilities and classical real-valued probabilities. Topractitioners of mathematics in Robinson’s framework it would be nat-ural to interpret such a “match” in terms of the constraint of being internal . By envisioning the possibility of introducing further con-straints, AP opens the door to introducing the internal condition. Theinternal condition would in any case be the obvious first choice for apractitioner of mathematics in Robinson’s framework. It is a constraintthat AP failed to consider. Similar shortcomings of Elga’s analysis in([17], 2004) were signaled by Herzberg ([21], 2007).In addition, it is possible to introduce further constraints. As an ex-ample, we propose two different hyperfinite representations of the uni-form spinners. Hyperfinite entities, being internal, satisfy the transferprinciple. Moreover, each of the following hyperfinite models will havevarious properties that improve upon
Tot , Un n , Reg , ¬ Sy R and Co of the measure P n .The main result of Benci et al. [3] entails that there exists • an algebra of Lebesgue measurable sets B ⊆ P ([0 , π )) suchthat for every A ∈ B , either A = ∅ or P µ ( A ) = 0; • a hyperfinite set Ω ⊆ ∗ [0 , π ) such that for every x ∈ [0 , π ), ∗ x ∈ Ω; • a hyperfinite probability measure P Ω over Ωwith the properties Tot , Un | Ω | , Reg , ¬ Sy R , Co and the followingadditional property: Un B P Ω is uniform over sets of B , i.e., P Ω ( ∗ A ∩ Ω) = P Ω ( ∗ B ∩ Ω)whenever
A, B ∈ B and P µ ( A ) = P µ ( B ). Internal sets are elements of ( ∗ -extensions of) classical sets; see [12], Section 3.1.Robinson’s book ([29], 1966) deals with internal entities systematically. An interesting and nontrivial choice is the ring of finite unions of intervals of theform [ a, b ). For more details, see Benci et al. ([4], 2015, p. 43).
NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 11
As a consequence of Un | Ω | , P Ω satisfies Un n for all n ∈ N . Moreover,the property Un B implies that P Ω is symmetric with respect to rota-tions by an arbitrary real angle, but only on the nonstandard extensionsof sets in B . For more details, we refer to [3] and [4, Section 3], wherethe example of the Lebesgue measure is discussed in detail.The hyperfinite measure P Ω can also be used to define a non-Archi-medean probability over the sample space [0 , π ) by setting(2.4) P ( A ) = P Ω ( ∗ A ∩ Ω) for each A ⊆ [0 , π ) . The hypothesis that for every x ∈ [0 , π ), ∗ x ∈ Ω ensures that P Ω ( x ) = | Ω | for every x ∈ [0 , π ). Thus the measure P of (2.4) is regular anduniform in the sense that it assigns the same nonzero probability tosingletons.In Section 3.2 we argue that non-Archimedean probabilities thatsatisfy the Omega-limit axiom of Benci et al. can be obtained in asimilar way, i.e., by restricting a suitable hyperfinite probability. Inthis external measure over [0 , π ), the choice for the probability of asingleton is uniquely determined from the underlying hyperfinite set Ω,and it is not arbitrary as argued by AP.By using an earlier result by Wattenberg [36], it is possible to obtainanother hyperfinite set Ω H and a hyperfinite probability measure P H over Ω H that has an additional advantage. Namely, it is coherentnot only with the Lebesgue measure, but also with the Hausdorff t -measures.Recall that the Hausdorff outer measure of order t of a set A ⊆ R isdefined as H t ( A ) = lim δ → (X n ∈ N λ ( I n ) t : A ⊆ [ n ∈ N I n , λ ( I n ) < δ ) . The Hausdorff measure H t is the restriction of the outer measure H t to the σ -algebra of measurable subsets of Ω.The probability measure P H over Ω H has the properties Tot , Un | Ω H | , Reg , ¬ Sy R as well as the following property: Co H if B is a Borel set with finite nonzero Hausdorff t -measure forsome real t ∈ [0 , + ∞ ), then for every H t -measurable set A ⊆ Ω H the sets A H = ∗ A ∩ Ω H and B H = ∗ B ∩ Ω H satisfy therelations H t ( A ∩ B ) H t ( B ) ≈ P H ( A H ∩ B H ) P H ( B H ) = P H ( A H | B H ) . Notice that, by taking t = 1 and B = [0 , π ), Co H implies Co . Each of the probability measures P n , P Ω and P H described abovemodels different aspects of the spinner. In particular, they are all uni-form, regular and total probability measures; moreover they approx-imate to varying degrees the properties of the real continuous modelbased upon the Lebesgue measure or upon the Hausdorff measure.Furthermore, the probability P Ω shows how the critique that non-Archimedean probabilities do not preserve intuitive symmetries, pre-sented by AP in [28, Section 3.3], can be addressed by means of asuitable hyperfinite model.Meanwhile, the probability P H shows that hyperreal-valued probabil-ity measures can be used simultaneously to represent the uncountablymany Hausdorff t -measures. The strength of these kinds of hyperfinitemodels is not discussed by AP, who only considers the coherence of anon-Archimedean measure with a single real-valued measure.3. Are infinitesimal probabilities underdetermined?
In Section 2 we presented an analysis of AP’s non-Archimedean mod-eling. Now we turn to his underdetermination theorems. AP’s firsttheorem is based upon the following construction. Let P be a prob-ability function with values in a non-Archimedean ordered extensionof R . AP sets(3.1) P α ( A ) = St( P ( A )) + α ( P ( A ) − St( P ( A )))for every α > AP’s first theorem expresses the fact that P α isinfinitely close to P and satisfies the same inequalities as P does. Thisresult can be interpreted as the fact that the probabilities P and P α induce the same comparisons between events.AP’s second theorem asserts a similar result for probability measuresthat satisfy the Ω-limit axiom of Benci et al. In this case, starting froma probability that satisfies the Ω-limit axiom (see Section 3.2) and froman automorphism φ of ∗ R that fixes only the standard real numbers, itis possible to define a new probability(3.2) P φ = φ ◦ P that yields the same comparisons between events as P , and that stillsatisfies the Ω-limit axiom.Both results are technically correct, but what AP fails to mention isthat, if P is internal, then the additional probabilities P α and P φ are allexternal whenever they differ from P , as we show in Sections 3.1 and AP assumes in addition that α ∈ R . However his results are still valid for everypositive finite α ∈ ∗ R . NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 13 parasitic in the sense of Clendinnen [14]. The significance of transfer and related principles both in the cur-rent practice of non-Archimedean mathematics based upon Robinson’sframework (see [12], Sections 3.1 and 3.2) and in the historical develop-ment of mathematical theories with infinitesimals (see [12], Section 3.3)is sufficient reason to recast AP’s theorems as a warning against the useof external probabilities in hyperreal modeling. Thus a careful analysisof Prussian theorems enables a meaningful criterion for the rejection ofAP’s underdetermination charge.For the purposes of the discussion that follows, recall that the trans-fer principle entails that any internal probability measure on a hyperfi-nite sample space Ω is hyperfinitely additive , i.e., that for every internal A ⊆ Ω, one has P ( A ) = P ω ∈ A P ( { ω } ). Consequently, if a probabilitymeasure is not hyperfinitely additive, then it is not internal.3.1. The first theorem of Pruss.
We begin our analysis with AP’sfirst theorem. AP is not clear on the domain of P ; here we will assumethat P is an internal probability measure defined over a hyperfiniteset Ω and that P ( { ω } ) ≈ ω ∈ Ω. This hypothesis is notrestrictive, since these measures are general enough to represent everynon-atomic real-valued probability measure [3, Theorem 2.2, p. 6].We define P α as in (3.1). Then the following theorem holds. Theorem 1.
Let Ω be a hyperfinite set and let P : ∗ P (Ω) → ∗ R be aninternal probability measure that satisfies P ( { ω } ) ≈ for all ω ∈ Ω .If α = 1 then P α is external. The significance of such externality can be appreciated in light ofthe following fact. If { X i : i < H } is an internal sequence of sets ofinfinite hyperfinite length H , and P an internal measure, then the sum P i P ( X i ) is well defined in ∗ R ; but if P is not internal then generallyspeaking P i P ( X i ) cannot be reasonably defined at all. Proof of Theorem 1.
Since P is internal, it is hyperfinitely additive. Asa consequence, X ω ∈ Ω P α ( { ω } ) = X ω ∈ Ω αP ( { ω } ) = α X ω ∈ Ω P ( { ω } ) = α. See note 1. A real-valued probability function is called non-atomic if and only if it assignsmeasure 0 to every singleton. If a real-valued probability measure P can be decom-posed into a sum of a non-atomic measure P na and a discrete measure P d , then ourTheorem 1 can still be applied to the hyperfinite representatives of P na . Since P α (Ω) = 1, the probability P α is hyperfinitely additive if andonly if α = 1, so that if α = 1 then P α cannot be hyperfinitely additiveand, as a consequence, it is external. (cid:3) The Ω -limit axiom. A similar argument refutes AP’s interpre-tation of his second theorem; see Section 3.3. Before showing that theprobability measures P φ obtained by AP are external, it will be con-venient to recall the Ω-limit axiom of Benci et al. [5] and some of itsconsequences. In this subsection, Ω will be a set of classical mathemat-ics. Define also Λ = { A ⊆ Ω : | A | ∈ N } . Thus if λ ∈ Λ, then λ is afinite set.The Ω-limit is a notion of limit governed by the following definition. Definition 1.
Let Ω be an infinite set and F an ordered field F ⊃ R .An Ω-limit in F is a correspondence that associates to every func-tion f : Λ → R , an element of F , denoted by lim λ ↑ Ω f , in such a waythat the following properties hold:(1) if there is a λ ∈ Λ with f ( λ ) = c ∈ R for every λ ⊇ λ ,then lim λ ↑ Ω f ( λ ) = c ;(2) for every f, g : Λ → R , one has • lim λ ↑ Ω ( f ( λ ) + g ( λ )) = lim λ ↑ Ω f ( λ ) + lim λ ↑ Ω g ( λ ) , and • lim λ ↑ Ω ( f ( λ ) · g ( λ )) = lim λ ↑ Ω f ( λ ) · lim λ ↑ Ω g ( λ ) . It is possible to obtain an Ω-limit by a suitable ultrapower construc-tion. If one defines F = R Λ / U , where U is a fine and free ultrafilterover Λ, then F is a field of hyperreal numbers. An Ω-limit over F can then be obtained by setting lim λ ↑ Ω f ( λ ) = [ f ] U .A probability function P : P (Ω) → ∗ R satisfies the Ω-limit axiom ifand only if there exists an Ω-limit such that P ( A ) = lim λ ↑ Ω P ( A | λ ) forevery A ⊆ Ω.Using the Ω-limit axiom it is possible to define a notion of infinitesum for P . Thus, for every A ⊆ Ω, the sum P ω ∈ A P ( ω ) is definedas lim λ ↑ Ω (cid:0)P ω ∈ A ∩ λ P ( ω ) (cid:1) ; see also [6, p. 6].Since a probability that satisfies the Ω-limit axiom is defined over aclassical set Ω, the non-Archimedean probabilities that satisfy the Ω-limit axiom are external [6, p. 25]. However, this is only due to thechoice by Benci et al. of working with a sample space that is not inter-nal. In fact, we will now show that non-Archimedean probabilities that An ultrafilter U over Λ is fine whenever for every λ ∈ Λ = { A ⊆ Ω : | A | ∈ N } , onehas { L ⊆ Λ : λ ∈ L } ∈ U . Such ultrafilters were referred to as adequate in earlierliterature; see e.g., Kanovei–Reeken [23, p. 143]. A free ultrafilter is an ultrafilterthat does not have a ⊆ -least element. NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 15 satisfy the Ω-limit axiom can be obtained as the restriction of suitableinternal hyperfinite probabilities. Let Ω ⊆ R . We define the set(3.3) Ω Λ = (cid:26) lim λ ↑ Ω f ( λ ) : Λ f → R and f ( λ ) ∈ λ (cid:27) . Then Ω Λ is an internal hyperfinite set that represents Ω. Moreover, itis possible to define an internal, uniform probability measure(3.4) P : ∗ P (Ω Λ ) → ∗ R by setting P ( A ) = | A || Ω Λ | for every internal A ⊆ Ω Λ . With this definition,one has P ( A ) = P ( ∗ A ∩ Ω Λ ) = | ∗ A ∩ Ω Λ || Ω Λ | for every A ⊆ Ω. A proof of these statements can be obtained fromthe proof of Theorem 2.2 in [3] as well as [6, Section 3.6]. Observe alsothat, if φ is an automorphism of ∗ R , then P φ ( A ) = φ (cid:0) P ( ∗ A ∩ Ω Λ ) (cid:1) .3.3. The second theorem of Pruss.
As we showed in Section 3.2,non-Archimedean probabilities that satisfy the Ω-limit are restrictionsof hyperfinite internal measures. We can now state our result con-cerning AP’s probabilities P φ . First, recall that in [28, Section 3.6,Proposition 1] AP proves that there are uncountably many nontrivialautomorphisms of ∗ R that fix R . However, these automorphisms areall external. Proposition 1. If φ : ∗ R → ∗ R is a field automorphism, then either φ is the identity or φ is external.Proof. It is well known that the only field automorphism of R is theidentity. Thus, by transfer, the only internal field automorphism of ∗ R is the identity. (cid:3) Our Proposition 1 already suggests that probability measures P φ areexternal whenever they do not coincide with P . However, we will provesuch a result explicitly, establishing our counterpart to AP’s secondtheorem. Theorem 2.
Let P : P (Ω) → ∗ R be a probability measure that satisfiesthe Ω -limit axiom and let Ω Λ and P be defined as in (3.3) and (3.4) .Suppose also that φ : ∗ R → ∗ R is a field automorphism. Then the mea-sure P φ = φ ◦ P is internal if and only if the restriction of φ to therange of P is the identity. This hypothesis can be avoided by extending the notion of Ω-limit as shown byBottazzi [8, Sections 4.2 and 4.3].
Proof.
Recall that the range of the measure P is the hyperfinite set S = (cid:8) , n , n , . . . , kn , . . . , (cid:9) , where n = | Ω Λ | . Let ψ be the restriction of φ to the range of P . If ψ is the identity, then P φ = P .Suppose ψ is an internal automorphism different from the iden-tity. Then the set n , ψ ( n ) , ψ ( n ) , . . . , ψ ( k ) ψ ( n ) , . . . , o is internal. Multi-plying every element of this set by ψ ( n ), we would obtain that theset { , , , . . . , ψ ( k ) , . . . , ψ ( n ) } is internal, as well. Let k be the leastnumber such that ψ ( k ) = k . Then ψ ( k −
1) = k −
1, so ψ ( k ) = ψ ( k − ψ ( k −
1) + ψ (1) = k − k , a contradiction.Thus ψ is necessarily external (or the identity). It remains to showthat ψ ◦ P is external. Since P is an internal mapping of the hyper-finite set ∗ P (Ω Λ ) onto S = (cid:8) , n , n , . . . , (cid:9) , it has an internal rightinverse χ : S → ∗ P (Ω Λ ) such that P ◦ χ is the identity on S . If ψ ◦ P were internal, then ψ ◦ P ◦ χ = ψ would be internal, contrary to ourhypothesis. (cid:3) Internal probability measures are not underdetermined.
Our Theorems 1 and 2 indicate that the underdetermination issueraised by AP is present only whenever one considers external prob-ability measures in addition to the internal ones. Significantly, suchexternal probability measures do not exist in Nelson’s Internal SetTheory (see [12], Section 3.2). Notice also that working with inter-nal probability measures is not restrictive, since, as we have shown,non-Archimedean functions that satisfy the Ω-limit can be obtained asthe restriction of suitable internal hyperfinite probabilities.It should be noted that we are not claiming that external mea-sures are not useful for hyperreal models. In fact, the external non-Archimedean probabilities by Benci et al. and the
Loeb measures , thatare obtained from suitable internal measures, play a relevant role forthe development of infinitesimal probabilities and for the hyperrealmeasure theory, respectively. What we showed is that the externalprobabilities proposed by AP do not have even basic properties such ashyperfinite additivity. As a consequence, these external probabilitiesare clearly inferior to their internal counterparts.In conclusion, once a hyperfinite sample space Ω is determined, thereis only one internal, uniform and regular probability measure P over Ω. Recall that the Loeb measure of an internal measure µ is obtained by compos-ing µ with the (external) standard part function. By the Caratheodory extensiontheorem, the resulting pre-measure is then extended to a measure defined on anexternal σ -algebra, that is usually called the Loeb measure associated to µ . NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 17
As a consequence, internal uniform probability measures with hyperfi-nite support are not underdetermined , contrary to what was alleged byPruss. Indeed, Pruss’ pair of theorems can be interpreted as a warningagainst the use of external measures in hyperreal probabilities. Asalready noted in Section 2.5, Pruss envisions the possibility of introduc-ing additional constraints. The discussion above suggests that the veryfirst constraint one should envision is the natural (and even obvious ,to a practitioner of mathematics in Robinson’s framework) constraintof being internal .In an analysis of the phenomenon of underdetermination, Clendinnenspeaks of “empirically equivalent alternatives [that are] parasitic on[the original] theory” [14, p. 63], an apt description of Pruss’ externalmeasures of the form (3.1) and (3.2).4.
Probability measures on transferless fields
Pruss claims that some extensions of the real numbers with infinites-imals, such as the surreal numbers, the fields of Laurent series or theLevi-Civita field, might provide a non-arbitrary choice of infinitesimalsfor the development of non-Archimedean probabilities.Due to the absence of a transfer principle, already discussed in [12,Section 3.4], there are some obstacles to developing a probability theoryover these non-Archimedean extensions of the real numbers. We willdiscuss two examples in detail in Sections 4.1 and 4.2.4.1.
Probability measures on the surreal numbers.
Pruss sug-gests that Conway’s surreal numbers (usually denoted No ) might besuitable for the development of infinitesimal probabilities, sinceWhatever you can do with hyperreals, you can do withsurreals, since any field of hyperreals can be embeddedin the surreals. [28, Section 2]Pruss seeks support for such a claim in Ehrlich [16]. In fact, the result[16, Theorem 20] entails that in the von Neumann–Bernays–G¨odel settheory with global choice, there exists a unique (up to an isomorphism)structure ( R , R ∗ , ∗ ) such that R ∗ satisfies Keisler’s axioms for hyperrealnumber systems and R ∗ is a proper class isomorphic to the class ofsurreal numbers. This argument and the possibility of uniquely specifying a hyperreal field, asdiscussed in Section 2.1 of [12], refute Pruss’ claim that hyperreal probabilities areunderdetermined both in the choice of a specific hyperreal field and in the choiceof the infinitesimal probability of singletons.
Ehrlich observes, furthermore, that every field of hyperreal numbersis isomorphic to an initial segment of the surreal numbers:Since every real-closed ordered field is isomorphic to aninitial subfield of No , the underlying ordered field ofany hyperreal number system is likewise isomorphic toan initial subfield of No . For example, the familiarultrapower construction of a hyperreal number systemas a quotient ring of R N (modulo a given nonprincipalultrafilter on N ) is isomorphic to No ( ω ) . . . assuming[the continuum hypothesis] . . . Similarly, if we assumethere is an uncountable inaccessible cardinal, ω α beingthe least, then No ( ω α ) . . . is isomorphic to the un-derlying ordered field in the hyperreal number systememployed by Keisler in his Foundations of InfinitesimalAnalysis . [16, Section 9, p. 35]Indeed, it is only the isomorphisms mentioned by Ehrlich that en-dow the surreal numbers (or their initial segments) with the transferprinciple of Robinson’s framework. In addition, the ∗ -map allowsfor the extension of functions in a way that the simplicity hierarchicalstructure of the surreal numbers is not yet able to handle [7].If one refrains from exploiting the transfer principle of Robinson’sframework, then the current developments of analysis over the surrealfield are fairly limited. For instance, it is still an open problem todefine an integral over No , despite some limited results obtained inthe last two decades (Costin et al. [15], 2015; Fornasiero [19], 2004).As a consequence, it is not currently possible to develop a measuretheory, let alone a probability theory, over the surreal numbers.4.2. Probability measures on the Levi-Civita field.
At the otherend of the spectrum, Pruss advocates the use of the “elegantly small”Levi-Civita field R . This extension of the real numbers was introducedby Levi-Civita in [24, 25]. It is the smallest non-Archimedean orderedfield extension of the field R of real numbers that is both real closedand complete in the order topology. For an introduction, we refer toLightstone–Robinson [26] and to Berz–Shamseddine [34]. It should be noted that the property of being an ordered field is a small fractionof the properties of the reals required to develop any substantial calculus. Thesurreals lack many relevant properties even with regard to their collection of naturalnumbers; see [12], note 28. As well as with the natural numbers and all the associated structure over h R ; N i necessary to develop calculus and measure theory. As opposed to other forms of completeness.
NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 19
The Levi-Civita field is defined as the set R = (cid:8) x ∈ R Q : ∀ q ∈ Q supp( x ) ∩ ( −∞ , q ] is finite (cid:9) together with the pointwise sum, and the product defined by the for-mula(4.1) ( x · y )( q ) = X q + q = q x ( q ) · y ( q ) . If one defines the element d ∈ R by posing d ( q ) = (cid:26) q = 10 if q = 1 , then every number in R can be written as a formal sum x = P q ∈ Q a q d q . In this sum the set Q ( x ) = { q ∈ Q : a q = 0 } has the property that theintersection Q ( x ) ∩ ( −∞ , q ] is finite for every q ∈ Q . Elements of R can be identified with those elements in R whose support is a subsetof the singleton { } . The field R can be linearly ordered and issequentially complete in the order topology, but due to the presenceof infinitesimal elements it is totally disconnected (the topologicalproperties of R are discussed in detail by Shamseddine [31]).If one considers the language L of ordered fields, then the realnumbers R , fields of hyperreal numbers and the Levi-Civita field R Pruss incorrectly states that only finitely many of the a q are nonzero [28, Sec-tion 2]. It is customary to define the order on R in a way that the number d is aninfinitesimal. This property is shared by every non-Archimedean extension of R . However,fields of hyperreal numbers of Robinson’s framework overcome this limitation byworking with internal sets and functions. For instance, in both the Levi-Civita fieldand in ∗ R the series P n ∈ N r is not defined unless r = 0, but a sum of hyperfinitelymany copies of r ∈ ∗ R is well-defined. Similarly, in the Levi-Civita field one shouldtake into account that the function f defined by f ( x ) = (cid:26) x ≈
00 if x R , but it does not satisfy e.g. the IntermediateValue Theorem or the Mean Value Theorem. Meanwhile, the counterpart of thisfunction in Robinson’s framework is external, so it does not provide a counterex-ample to the Intermediate Value Theorem or to the Mean Value Theorem. Namely L = { + , − , , · , − , , < } , where + and · are binary functions, − and − are unary functions, 0 and 1 are constant symbols, and < is a binary relation. are L -structures which are models of the model-complete L -theoryof real closed ordered fields [13, 27]. However, the Levi-Civita field isnot elementarily equivalent to R , and the only result analogous to thetransfer principle between R and R is fairly limited. In particular, onlylocally analytic functions can be extended from real closed intervals toclosed intervals in R in a way that preserves elementary properties [9,Sections 3, 4].Consider now the uniform measure m on the Levi-Civita field studiedby Berz–Shamseddine ([33], 2003), Shamseddine ([32], 2012), and Bot-tazzi ([10], 2020). A set is measurable with respect to m if it can be ap-proximated arbitrarily well by intervals, in analogy with the Lebesguemeasure. However, due to the properties of the topology of the Levi-Civita field, the measure m turns out to be rather different from theLebesgue measure over R . For instance, in R the complement of ameasurable set is not necessarily measurable. As a consequence, thefamily of measurable sets is not even an algebra.As with the Lebesgue measure, the family of measurable functions isobtained from a family of simple functions. For the Lebesgue measure,these are the step functions; however a similar choice in the Levi-Civitafield would lead to a very narrow class of measurable functions. A morefruitful choice for the Levi-Civita field is to define the simple functionsas the analytic functions over a closed interval.Regardless of the choice of simple functions, Shamseddine and Berzproved that any measurable function on a set A ⊆ R is locally simplealmost everywhere on A [33, Proposition 3.4, p. 379]. As a consequence,an absolutely continuous probability measure over R can only have alocally analytic density function; similarly, the only real probabilityfunctions that can be extended to probability functions over the Levi-Civita field are locally analytic. A partial converse of this negativeresult shows that measurable functions are not expressive enough toapproximate with an infinitesimal error a real probability that is notlocally analytic at any point of its domain [10, Proposition 3.16].We now turn our attention to discrete probability measures definedover the Levi-Civita field. We note that this field has no notion ofhyperfiniteness or of sum over sets that are not countable. Moreover,if h ∈ R is an infinitesimal, the limit lim n →∞ P ni =0 h = lim n →∞ nh isnot defined, since the sequence { nh } n ∈ N does not converge in R . Thisis a significant difference with hyperreal-valued measures, where one A theory in a language L is model-complete if for every pair M and N of modelsof the theory, if there is an embedding i : M ֒ → N , then the embedding is elemen-tary. An embedding is elementary whenever for every first-order formula φ in thelanguage L , M | = φ ( a , . . . , a n ) if and only if N | = φ ( i ( a ) , . . . , i ( a n ). NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 21 can sum a internal hyperfinite family of infinitesimal terms and obtaina well-defined hyperreal result.As a consequence, whenever Ω ⊆ R is not a finite set, there can-not be a regular probability measure m over Ω that assigns the sameprobability to every ω ∈ Ω. In this regard, R -valued discrete probabil-ity measures do not offer any improvement upon real-valued discreteprobability measures.These properties of the uniform measure and of the discrete prob-ability measures over the Levi-Civita field impose serious limitationson the use of the Levi-Civita field as the target space of a probabilitymeasure, be it absolutely continuous, discrete, or a combination of thetwo.4.3. Probability measures in other fields with infinitesimals.
The Hahn field, obtained from functions x ∈ R Q with well-orderedsupport with the pointwise sum and the product defined as in for-mula (4.1), shares some properties with the Levi-Civita field. Never-theless, there is no measure theory over the Hahn field yet, even thoughsome preliminary results on the convergence of power series were ob-tained by Flynn and Shamseddine in ([18], 2019).Kaiser recently developed a uniform measure over a class of non-Archimedean real closed fields ([22], 2018). However, this measureis only defined for semialgebraic sets. This condition is even morerestrictive than the one imposed on the measurable sets in the Levi-Civita field (see Section 4.2). Note also that the resulting measure isonly finitely additive.The picture that emerges is very different from the one sketched byPruss. Robinson’s hyperreal fields have many advantages over othernon-Archimedean extensions of R , as already realized by Fraenkel (see[12], Section 3.3), and it is unlikely that the gap between these theorieswill be closed any time soon.5. Conclusion
In his
Synthese paper [28], Pruss claims that the infinitesimal prob-abilities of Robinson’s framework for analysis with infinitesimals are underdetermined . His claim hinges upon(1) some models of infinitesimal probabilities, and(2) a pair of theorems that entail the existence of uncountably manyinfinitesimal probabilities that yield the same decision-theoreticcomparisons as the original one.
In Section 2, we addressed issue (1) by showing that proper hyperfinitemodels avoid the underdetermination problem. In Section 3 we focusedon Prussian theorems mentioned in item (2), and proved that all of theadditional infinitesimal probabilities obtained by Pruss are external. Asa consequence, once a hyperfinite sample space has been chosen, there isonly one internal probability measure over it. The results of Sections 2and 3 suggest that the underdetermination critique of Pruss is limitedto external entities. Thus, working with internal sets and functionsin Robinson’s framework dissolves the underdetermination objection.While recognizing that it may be possible to narrow down the choice ofinfinitesimal probability using additional constraints, Pruss fails to con-sider the natural internal constraint. Pruss could respond by arguingthat internality is not the kind of criterion he had in mind; however, thepoint remains that internality is such an obvious choice that it shouldhave been addressed one way or another. The fact that the issue is notexamined in his paper constitutes a serious shortcoming of his analysis.A would-be critic of Robinson’s framework could then retreat to aneven more limited objection to the effect that the choice of the samplespace is underdetermined. However, in the discussion of the hyperfinitemodels we showed the following: • this choice is underdetermined in the Archimedean case, as well;however, this underdetermination is not problematic, since dif-ferent models (be they Archimedean or hyperfinite) are com-patible, as discussed in Sections 2.2 and 2.3. • it is also possible to specify additional criteria for the choiceof a hyperfinite model: for instance, it is possible to preserverotational symmetry on a non-trivial family of sets or coher-ence with uncountably many real-valued measures, as shown inSection 2.5. Thus the possibility of working with hyperfinitemodels that improve upon the properties of the Archimedeanones should be regarded as an advantage of hyperreal probabil-ities.In his critique, Pruss also suggests that other fields with infinitesimalsare more suitable for the development of infinitesimal probabilities.However, in [12], Sections 3.3 and 3.4 we showed that this claim ig-nores the Klein–Fraenkel criteria for the usefulness of a theory withinfinitesimals. In addition, in Section 4 we showed that probabilities We remark that, while the probability functions that satisfy the Ω-limit axiomare external, they are obtained as the restriction of suitable internal hyperfinitefunctions, as shown in Section 3.2. Thus they are also impervious to Pruss’ unde-termination claim.
NTERNALITY, TRANSFER, AND INFINITESIMAL MODELING 23 taking values in the surreal numbers and in the Levi-Civita field are lessexpressive than real-valued probabilities, mainly due to the absence ofa transfer principle for these structures. Moreover, the absence of acomprehensive transfer principle makes such fields vulnerable to The-orem 1 of Pruss. In contrast to hyperreal fields, transferless fields donot possess a notion of internality and are thus unable to escape un-derdetermination.
Acknowledgments
We are grateful to Karel Hrbacek and Vladimir Kanovei for insightfulcomments on earlier versions that helped improve our article, and toanonymous referees for constructive criticism. The influence of HiltonKramer (1928–2012) is obvious.
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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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