An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals
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
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 ). Does this apply to infinitesimals, as well?
The exposition in this section follows R. Arthan . 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
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 . 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.
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
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 ).
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  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 , 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, ) who published three years earlier than E. Heine and Cantor. can be represented schematically by the diagram
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
where is as above, alludes to the set-theoretic power operation, and selects his cuts. For a history of the problem, see P. Ehrlich .
An alternative approach was proposed by Schanuel, and developed by N. A’Campo , R. Arthan , ,555Arthan’s Irrational construction of from  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 , R. Street , O. Deiser [15, pp. 112-127], and others, who follow the formally simpler blueprint
where selects certain almost homomorphisms from to itself, such as the map
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 :
where identifies sequences which differ on a finite set of indices:
Here the constant sequences induce an inclusion
Such a construction is closely related to (a version of) the -calculus of Schmieden and Laugwitz . 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.
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
where is a fixed ultrafilter on , and add negatives. The resulting modification of (3.5), called an ultrapower, will be denoted
and is related to Skolem’s construction in 1934 of a nonstandard model of arithmetic . 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 ).
The traditional ultrapower approach to constructing the hyperreals is to start with the field of real numbers and build the ultrapower
however, this construction employs needless intermediate procedures as described above. Our approach is to follow instead the “skip the rationals” blueprint
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
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
generates an infinitesimal in IIR since the almost homomorphisms are “getting flatter and flatter” for large time .
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 (, , , , ) to the Bolzmann equation (see L. Arkeryd [4, 5]); modeling of timed systems in computer science (see H. Rust ); mathematical economics (see R. Anderson ); mathematical physics (see Albeverio et al. ); etc. A comprehensive re-appraisal of the historical antecedents of modern infinitesimals has been undertaken in recent work by Błaszczyk et al. , Borovik et al. , Bråting , Kanovei , Katz & Katz [31, 29, 32, 30], Katz & Leichtnam , 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. .
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.
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 .
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
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 ) 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]).
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
Notice that is a subset of . Hence can be viewed as a subset of . Again for each
Then is an isomorphism from
to . Therefore, as an elementary subfield of is isomorphic to the hyperreal field .
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 .
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.
is a class hyperreal field, and any definable hyperreal field admits an elementary imbedding into .
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
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 . ∎
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 . By the same idea of constructing above we can follow the steps in  or  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 . 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.
We are grateful to R. Arthan, P. Ehrlich, and V. Kanovei for helpful comments.
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