An integer construction of infinitesimals

An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals

Alexandre Borovik School of Mathematics, University of Manchester, Oxford Street, Manchester, M13 9PL, United Kingdom alexandre.borovik@manchester.ac.uk Renling Jin Department of Mathematics, College of Charleston, South Carolina 29424, USA JinR@cofc.edu  and  Mikhail G. Katz Department of Mathematics, Bar Ilan University, Ramat Gan 52900 Israel katzmik@macs.biu.ac.il
Abstract.

A construction of the real number system based on almost homomorphisms of the integers  was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).

Key words and phrases:
Eudoxus; hyperreals; infinitesimals; limit ultrapower; universal hyperreal field
2000 Mathematics Subject Classification:
Primary 26E35; Secondary 03C20
00footnotetext: Supported by the Israel Science Foundation grant 1517/12

1. From Kronecker to Schanuel

Kronecker famously remarked that, once we have the natural numbers in hand, “everything else is the work of man” (see Weber [49]). Does this apply to infinitesimals, as well?

The exposition in this section follows R. Arthan [7]. A function  from  to  is said to be an almost homomorphism if and only if the function  from  to  defined by

has bounded (i.e., finite) range, so that for a suitable integer , we have  for all . The set denoted

(1.1)

of all functions from  to  becomes an abelian group if we add and negate functions pointwise:

It is easily checked that if  and  are almost homomorphisms then so are  and . Let  be the set of all almost homomorphisms from  to . Then  is a subgroup of . Let us write  for the set of all functions from  to  whose range is bounded. Then  is a subgroup of . The “Eudoxus reals” are defined as follows.111The term “Eudoxus reals” has gained some currency in the literature, see e.g., Arthan [7]. Shenitzer [46, p. 45] argues that Eudoxus anticipated 19th century constructions of the real numbers. The attribution of such ideas to Eudoxus, based on an interpretation involving Eudoxus, Euclid, and Book 5 of The Elements, may be historically questionable.

Definition 1.1.

The abelian group  of Eudoxus reals is the quotient group .

Elements of  are equivalence classes,  say, where  is an almost homomorphism from  to , i.e.,  is a function from to  such that  defines a function from  to  whose range is bounded. We have if and only if the difference  has bounded range, i.e., if and only if  for some  and all  in .

The addition and additive inverse in  are induced by the pointwise addition and inverse of representative almost homomorphisms:

where  and  are defined by

and

for all  in .

The group  of Eudoxus reals becomes an ordered abelian group if we take the set  of positive elements to be

The multiplication on  is induced by composition of almost homomorphisms. The multiplication turns  into a commutative ring with unit. Moreover, this ring is a field. Even more surprisingly,  is an ordered field with respect to the ordering defined by .

Theorem 1.2 (See Arthan [7]).

is a complete ordered field and is therefore isomorphic to the field of real numbers .

The isomorphism  assigns to every real number , the class of the function

where  is the integer part function.

In the remainder of the paper, we combine the above one-step construction of the reals with the ultrapower or limit ultrapower construction, to obtain hyperreal number systems directly out of the integers. We show that any hyperreal field, whose universe is a set, can be so obtained by such a one-step construction. Following this, working in NBG (von Neumann-Bernays-Gödel set theory with the Axiom of Global Choice), we further observe that by using a suitable definable ultrapower, even the maximal (i.e., the On-saturated)222Recall that a model is On-saturated if is -saturated for any cardinal in On. Here On (or ON) is the class of all ordinals (cf. Kunen [40, p. 17]). A hyperreal number system is On-saturated if it satisfies the following condition: If is a set of equations and inequalities involving real functions, hyperreal constants and variables, then has a hyperreal solution whenever every finite subset of has a hyperreal solution (see Ehrlich [17, section 9, p. 34]). hyperreal number system recently described by Ehrlich [17] can be obtained in a one-step fashion directly from the integers.333Another version of such an On-saturated number system was introduced by Kanovei and Reeken (2004, [25, Theorem 4.1.10(i)]) in the framework of axiomatic nonstandard analysis. As Ehrlich [17, Theorem 20] showed, the ordered field underlying an On-saturated hyperreal field is isomorphic to J. H. Conway’s ordered field No, an ordered field Ehrlich describes as the absolute arithmetic continuum (modulo NBG).

2. Passing it through an ultraproduct

Let now  be the ring of integer sequences with operations of componentwise addition and multiplication. We define a rescaling to be a sequence  of almost homomorphisms . Rescalings are thought of as acting on , hence the name. A rescaling  is called bounded if each of its components, , is bounded.

Rescalings factorized modulo bounded rescalings form a commutative ring  with respect to addition and composition. Quotients of  by its maximal ideals are hyperreal fields. Thus, hyperreal fields are factor fields of the ring of rescalings of integer sequences. This description is a tautological translation of the classical construction, due to E. Hewitt [23], but it is interesting for the sheer economy of the language used. We will give further details in the sections below.

3. Cantor, Dedekind, and Schanuel

The strategy of Cantor’s construction of the real numbers444The construction of the real numbers as equivalence classes of Cauchy sequences of rationals, usually attributed to Cantor, is actually due to H. Méray (1869, [42]) who published three years earlier than E. Heine and Cantor. can be represented schematically by the diagram

(3.1)

where the subscript  evokes the passage from a pair of integers to a rational number; the arrow alludes to forming sequences; and subscript  reminds us to select Cauchy sequences modulo equivalence. Meanwhile, Dedekind proceeds according to the scheme

(3.2)

where  is as above,  alludes to the set-theoretic power operation, and  selects his cuts. For a history of the problem, see P. Ehrlich [16].

An alternative approach was proposed by Schanuel, and developed by N. A’Campo [1], R. Arthan [6], [7],555Arthan’s Irrational construction of from [6] describes a different way of skipping the rationals, based on the observation that the Dedekind construction can take as its starting point any Archimedean densely ordered commutative group. The construction delivers a completion of the group, and one can define multiplication by analyzing its order-preserving endomorphisms. Arthan uses the additive group of the ring , which can be viewed as with an ordering defined using a certain recurrence relation. T. Grundhöfer [22], R. Street [48], O. Deiser [15, pp. 112-127], and others, who follow the formally simpler blueprint

(3.3)

where  selects certain almost homomorphisms from  to itself, such as the map

(3.4)

for real , modulo equivalence (think of  as the “large-scale slope” of the map).666One could also represent a real by a string based on its decimal expansion, but the addition in such a presentation is highly nontrivial due to carry-over, which can be arbitrarily long. In contrast, the addition of almost homomorphisms is term-by-term. Multiplication on the reals is induced by composition in , see formula (1.1). Such a construction has been referred to as the Eudoxus reals.777See footnote 1 for a discussion of the term. The construction of  from  by means of almost homomorphisms has been described as “skipping the rationals ”.

We will refer to the arrow in (3.3) as the space dimension, so to distinguish it from the time dimension occurring in the following construction of an extension of :

(3.5)

where  identifies sequences which differ on a finite set of indices:

(3.6)

Here the constant sequences induce an inclusion

Such a construction is closely related to (a version of) the -calculus of Schmieden and Laugwitz [45]. The resulting semiring has zero divisors. To obtain a model which satisfies the first-order Peano axioms, we need to quotient it further. Note that up to this point the construction has not used any nonconstructive foundational material such as the axiom of choice or the weaker axiom of the existence of nonprincipal ultrafilters.

4. Constructing an infinitesimal-enriched continuum

The traditional ultrapower construction of the hyperreals proceeds according to the blueprint

where  is a fixed ultrafilter on . Replacing  by any of the possible constructions of  from , one in principle obtains what can be viewed as a direct construction of the hyperreals out of the integers . Formally, the most economical construction of this sort passes via the Eudoxus reals.

An infinitesimal-enriched continuum can be visualized by means of an infinite-magnification microscope as in Figure 1.

Figure 1. Zooming in on infinitesimal 

To construct such an infinitesimal-enriched field, we have to deal with the problem that the semiring constructed in the previous section contains zero divisors.

To eliminate the zero divisors, we need to quotient the ring further. This is done by extending the equivalence relation by means of a maximal ideal defined in terms of an ultrafilter. Thus, we extend the relation defined by (3.6) to the relation declaring  and  equivalent if

(4.1)

where  is a fixed ultrafilter on , and add negatives. The resulting modification of (3.5), called an ultrapower, will be denoted

(4.2)

and is related to Skolem’s construction in 1934 of a nonstandard model of arithmetic [47]. We refer to the arrow in (4.2) as time to allude to the fact that a sequence that increases without bound for large time will generate an infinite “natural” number in IIIN. A “continuous” version of the ultrapower construction was exploited by Hewitt in constructing his hyperreal fields in 1948 (see [23]).

The traditional ultrapower approach to constructing the hyperreals is to start with the field of real numbers  and build the ultrapower

(4.3)

where the subscript  is equivalent to that of (4.2), see e.g. Goldblatt  [21]. For instance, relative to Cantor’s procedure (3.1), this construction can be represented by the scheme

however, this construction employs needless intermediate procedures as described above. Our approach is to follow instead the “skip the rationals” blueprint

(4.4)

where the image of each  is the sequence  with general term , so that . Thus a general element of IIR  is generated (represented) by the sequence

(4.5)

Here one requires that for each fixed element  of the exponent copy of , the map

is an almost homomorphism (space direction), while  in (4.4) alludes to the ultrapower quotient in the time-direction . For instance, we can use almost homomorphisms of type (3.4) with . Then the sequence

(4.6)

generates an infinitesimal in IIR  since the almost homomorphisms are “getting flatter and flatter” for large time .

Theorem 4.1.

Relative to the construction (4.4), we have a natural inclusion . Furthermore, IIR is isomorphic to the model of the hyperreals obtained by quotienting  by the chosen ultrafilter, as in (4.3).

Proof.

Given a real number , we choose the constant sequence given by  (the sequence is constant in time ). Sending  to the element of IIR  defined by the sequence

yields the required inclusion . The isomorphism  is obtained by letting

for each , and sending the element of IIR  represented by (4.5) to the hyperreal represented by the sequence . ∎

Denoting by  the infinitesimal generated by the integer object (4.6), we can then define the derivative of  at  following Robinson as the real number  infinitely close (or, in Fermat’s terminology, adequal)999See A. Weil [50, p. 1146]. to the infinitesimal ratio .

Applications of infinitesimal-enriched continua range from aid in teaching calculus ([18], [27], [28], [37], [43]) to the Bolzmann equation (see L. Arkeryd [4, 5]); modeling of timed systems in computer science (see H. Rust [44]); mathematical economics (see R. Anderson [3]); mathematical physics (see Albeverio et al. [2]); etc. A comprehensive re-appraisal of the historical antecedents of modern infinitesimals has been undertaken in recent work by Błaszczyk et al. [11], Borovik et al. [12], Bråting [13], Kanovei [24], Katz & Katz [31, 29, 32, 30], Katz & Leichtnam [33], Katz and Sherry [35, 36], and others. A construction of infinitesimals by “splitting” Cantor’s construction of the reals is presented in Giordano et al. [20].

5. Formalization

In this and the next sections we formalize and generalize the arguments in the previous sections. We show that by a one-step construction from -valued functions we can obtain any given (set) hyperreal field. We can even obtain a universal hyperreal field which contains an isomorphic copy of every hyperreal field, by a one-step construction from -valued functions.

We assume that the reader is familiar with some basic concepts of model theory. Consult [14] or [38] for concepts and notations undefined here.

Let and be two models in a language with base sets and , respectively. The model is called an -elementary extension of the model , or is an -elementary submodel of , if there is an embedding , called an -elementary embedding, such that for any first-order -sentence with parameters , is true in if and only if is true in .

Let and be two models in language with base sets and , respectively. Let

i.e., is formed by adding to an -dimensional relational symbol for each -dimensional relation on for any positive integer . Let be the natural -model with base set , i.e., the interpretation of in for each is . The model is called a complete elementary extension of if can be expanded to an -model with base set such that is an -elementary extension of .

It is a well-known fact that if is an ultrapower of or a limit ultrapower of , then is a complete elementary extension of .

In this section we always view the set as the set of all Eudoxus reals.

An ordered field is called a hyperreal field if it is a proper complete elementary extension of . Let

where is the collection of all finite-dimensional relations on . We do not distinguish between and the -model . By saying that is a hyperreal field we will sometimes mean that is the base set of the hyperreal field, but at other times we mean that  is the hyperreal field viewed as an -model. We will spell out the distinction when it becomes necessary.

Recall that is the set of all bounded functions from to . For a pair of almost homomorphisms , we will write if and only if . Let be an infinite set. If is a two-variable function from to and is a fixed element in , we write for the one-variable function from to .

Definition 5.1.

Let be any infinite set. We set

Let be a fixed non-principal ultrafilter on . For a pair of functions for some set , we set

Let .

Definition 5.2.

For any we will write

It is easy to check that is an equivalence relation on . For each let

For each we can consider as a function of from to . Thus the map

such that is an isomorphism from to . Hence can be viewed as an ultrapower of . Therefore, the quotient

is a hyperreal field constructed in one step from the set of -valued functions .

If the set is , then is exactly the hyperreal field IIR mentioned in the previous sections. Since can be any infinite set, we can construct a hyperreal field of arbitrarily large cardinality in one step from a set of -valued functions .

6. Limit ultrapowers and definable ultrapowers

If we consider a limit ultrapower instead of an ultrapower, we can obtain any (set) hyperreal field by a one-step construction from a set of -valued functions . The reader could consult (Keisler [38]) for the notations, definitions, and basic facts about limit ultrapowers. The main fact we need here is the following theorem (see Keisler [38, Theorem 3.7]).

Theorem 6.1.

If and are two models of the same language, then  is a complete elementary extension of if and only if is a limit ultrapower of .

Given any (set) hyperreal field , let be the ultrafilter on an infinite set and let be the filter on such that is isomorphic to the limit ultrapower . We can describe the limit ultrapower in one step from the set of -valued functions . For each , let

Let

Notice that is a subset of . Hence can be viewed as a subset of . Again for each

let

Then is an isomorphism from

to . Therefore, as an elementary subfield of is isomorphic to the hyperreal field .

Theorem 6.2.

An isomorphic copy of any (set) hyperreal field can be obtained by a one-step construction from a set of -valued functions for some filter on .

7. Universal and On-saturated Hyperreal Number Systems

We call a hyperreal field IIR universal if any hyperreal field, which is a set or a proper class of NBG, can be elementarily embedded in IIR. Obviously a universal hyperreal field is necessarily a proper class. We now want to construct a definable hyperreal field with the property that any definable hyperreal field that can be obtained in NBG by a one-step construction from a collection of -valued functions can be elementarily embedded in it. In a subsequent remark we point out that in NBG we can actually construct a definable hyperreal field so that every hyperreal field (definable or non-definable) can be elementarily embedded in it. Moreover, the universal hyperreal field so constructed is isomorphic to the On-saturated hyperreal field described in [17].

Notice that NBG implies that there is a well order on where is the class of all sets. A class is called -definable if there is a first-order formula with set parameters in the language such that for any set , if and only if is true in . Trivially, every set is -definable. We work within a model of NBG with set universe plus all -definable proper subclasses of . By saying that a class is definable we mean that is -definable.

Let be the class of all finite sets of ordinals, i.e., . Notice that is a definable proper class. Let be the collection of all definable subclasses of . Notice that we can code by a definable class.101010 This is true because each definable subclass of can be effectively coded by the Gödel number of a first-order formula in the language of and a set in . By the well order of we can determine a unique code for each definable class in . Using the global choice, we can form a non-principal definable ultrafilter such that for each , the definable class

is in . Again can be coded by a definable class. Let be the collection of all definable class functions from to such that for each , is an almost homomorphism from to . For any two functions and in , we write if and only if the definable class

is in . Let be the collection of all definable classes in such that . Then is an equivalence relation on . Let

By the arguments employed before, we can show that is isomorphic to the definable ultrapower of modulo . Hence is a complete elementary extension of . Therefore, is a hyperreal field. By slightly modifying the proof of [14, Theorem 4.3.12, p. 255] we can prove the following theorem.

Theorem 7.1.

is a class hyperreal field, and any definable hyperreal field admits an elementary imbedding into .

Proof.

We only need to prove the second part of the theorem. For notational convenience we view as instead of in this proof.

Given a definable hyperreal field , recall that is the language of ordered fields, and

Let be all quantifier-free -sentences with parameters such that is true in . Since is a definable class, is a definable class (under a proper coding). Let be the size of , i.e., is the cardinality of if is a set and if is a definable proper class. Let be a definable bijection from to .

For each we need to find a definable function such that the map is an -elementary embedding. We define these simultaneously.

Let . If let . Suppose and let . Notice that if , then . Let . Since

is true in , it is also true in . Let be the -least -tuple such that is true in . If , let . If for , then let . Since is quantifier-free, the functions are definable classes in NBG.

We now verify that such that is an -elementary embedding.

Let be an arbitrary -sentence with parameters
.

Suppose that is true in . Since defines an -ary relation on , we have that the -sentence

is true in . Hence is true in and in . One of the consequences of this is that is true in , hence it is in . Let be such that . If , then

Hence is true in by the definition of the ’s. Since is true in , we have that is true in . Thus

Since is a member of , we have that is true in .

Suppose that is false in . Then is true in . Hence by the same argument we have that

which implies that

Hence is an -elementary embedding from to . ∎

Remark 7.2.

We have shown that every definable hyperreal field can be elementarily embedded into . If we want to show that is universal, we need to elementarily embed every (definable or non-definable) hyperreal field into the definable hyperreal field . Notice that there are models of NBG with non-definable classes. The proof of Theorem 7.1 may not work when is a non-definable class because the bijection may not be definable and may not be definable. If is not definable, may not be an element in .

The idea of making every hyperreal field embeddable into is that we can make On-saturated by selecting a definable ultrafilter more carefully. Notice that NBG implies that every proper class has the same size On. Hence can be expressed as the union of On-many sets. If we can make sure that is On-saturated, i.e., -saturated for any set cardinality , then can be elementarily embedded into  although such an embedding may be non-definable.

The ultrafilter used in the construction of in the proof of Theorem 7.1 is a definable version of a regular ultrafilter. In order to make sure that is On-saturated, we need to require that be a special kind of definable regular ultrafilter called a definable good ultrafilter. The definition of a (set) good ultrafilter can be found in [14, p. 386]. The construction of an -good ultrafilter can be found in either [14, Theorem 6.1.4] or Kunen [39]. By the same idea of constructing above we can follow the steps in [39] or [14] to construct a definable class good ultrafilter on On. Now the ultrapower of modulo the definable class good ultrafilter is On-saturated. The proof of this fact is similar to that in [14]. However, since the definition of a definable class good ultrafilter and the proof of the saturation property of the ultrapower modulo a definable class good ultrafilter are long and technical, and the ideas are similar to what we have already presented above, we will not include them in this paper.

Another way of constructing a definable On-saturated hyperreal field is by taking the union of an On-long definable elementary chain of set hyperreal fields with the property that is -saturated. However, this construction cannot be easily translated into a “one-step” construction. Moreover, if we allow higher-order classes, we can express as a one-step limit ultrapower following the same idea as in the proof of [14, Theorem 6.4.10]. However, this is not possible in NBG since all classes allowed in a model of NBG are subclasses of . On the other hand, as we indicated above, the process of constructing as a definable ultrapower can be carried out in NBG, and done so in a “one-step” fashion.

Acknowledgments

We are grateful to R. Arthan, P. Ehrlich, and V. Kanovei for helpful comments.

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