Completeness of the Leibniz Field and Rigorousness of Infinitesimal Calculus
aa r X i v : . [ m a t h . HO ] A ug Completeness of the Leibniz Field andRigorousness of Infinitesimal Calculus
James F. Hall & Todor D. TodorovMathematics DepartmentCalifornia Polytechnic State UniversitySan Luis Obispo, California 93407, USA([email protected] & [email protected])
Abstract
We present a characterization of the completeness of the fieldof real numbers in the form of a collection of ten equivalent state-ments borrowed from algebra, real analysis, general topology andnon-standard analysis. We also discuss the completeness of non-Archimedean fields and present several examples of such fields. Asan application we exploit one of our results to argue that the Leibnizinfinitesimal calculus in the 18 th century was already a rigorous branchof mathematics – at least much more rigorous than most contempo-rary mathematicians prefer to believe. By advocating our particularhistorical point of view, we hope to provoke a discussion on the im-portance of mathematical rigor in mathematics and science in general.We believe that our article will be of interest for those readers whoteach courses on abstract algebra, real analysis, general topology, logicand the history of mathematics. Mathematics Subject Classification:
Primary: 03H05; Secondary: 01A50,03C20, 12J10, 12J15, 12J25, 26A06.
Key words:
Ordered field, complete field, infinitesimals, infinitesimal cal-culus, non-standard analysis, valuation field, power series, Hahn field, trans-fer principle. 1
Introduction
In Section 2 we recall the basic definitions and results about totally orderedfields – not necessarily Archimedean. The content of this section, althoughalgebraic and elementary in nature, is very rarely part of standard mathemat-ical education. In Section 3 we present a long list of definitions of differentforms of completeness of an ordered field. These definitions are not new,but they are usually spread throughout the literature of various branches ofmathematics and presented at different levels of accessibility. In Section 4 wepresent a characterization of the completeness of an Archimedean field – thatis to say, a characterization of the completeness of the reals. This character-ization is in the form of a collection of ten equivalent statements borrowedfrom algebra, real analysis, general topology and non-standard analysis (The-orem 4.1). Some parts of this collection are well-known or can be found inthe literature. We, believe however, that this is the first time that the wholecollection has appeared together; we also entertain the possibility that thiscollection is comprehensive. In Section 5 we establish some basic resultsabout the completeness of non-Archimedean fields which cannot be found ina typical textbook on algebra or analysis. In Section 6 we present numer-ous examples of non-Archimedean fields and discuss their completeness. Themain purpose of this section is to emphasize the essential difference in thecompleteness of Archimedean and non-Archimedean fields.In Section 7 we offer a short survey of the history of infinitesimal calculus,written in a polemic-like style . One of the characterizations of the complete-ness of an Archimedean field presented in Theorem 4.1 is formulated in termsof infinitesimals, and thus has a strong analog in the 18 th century infinites-imal calculus. We exploit this fact, along with some older results due to J.Keisler [12], in order to re-evaluate “with fresh eyes” the rigorousness of theinfinitesimal calculus. In Section 8 we present a new, and perhaps surpris-ing for many, answer to the question “How rigorous was the infinitesimalcalculus in the 18 th century?” arguing that the Leibniz-Euler infinitesimalcalculus was, in fact, much more rigorous than most todays mathematiciansprefer to believe. It seems perhaps, a little strange that it took more that200 years to realize how the 18 th century mathematicians prefer to phrasethe completeness of the reals. But better late (in this case, very late), thennever.The article establishes a connection between different topics from algebra,analysis, general topology, foundations and history of mathematics which2arely appear together in the literature. Section 6 alone – with the helpperhaps, of Section 2 – might be described as “the shortest introduction tonon-standard analysis ever written” and for some readers might be an “eyeopener.” We believe that this article will be of interest for all who teachcourses on abstract algebra, real analysis and general topology. Some partsof the article might be accessible even for advanced students under a teacher’ssupervision and suitable for senior projects on real analysis, general topologyor history of mathematics. We hope that the historical point of view whichwe advocate here might stir dormant mathematical passions resulting in afruitful discussion on the importance of mathematical rigor to mathematicsand science in general. In this section we recall the main definitions and properties of totally orderedrings and fields (or simply, ordered rings and fields for short), which are notnecessarily Archimedean. We shortly discuss the properties of infinitesimal,finite and infinitely large elements of a totally ordered field. For more details,and for the missing proofs, we refer the reader to (Lang [15], Chapter XI),(van der Waerden [31], Chapter 11) and (Ribenboim [24]). (Orderable Ring) . Let K be a ring (field). Then: K is called orderable if there exists a non-empty set K + ⊂ K such that:(a) 0 K + ; (b) K + is closed under the addition and multiplication in K ;(c) For every x ∈ K exactly one of the following holds: x = 0 , x ∈ K + or − x ∈ K + . K is formally real if, for every n ∈ N , the equation P nk =0 x k = 0 in K n admits only the trivial solution x = · · · = x n = 0. A field K is orderable if and only if K is formally real.Proof. We refer the reader to (van der Waerden [31], Chapter 11).The fields, Q and R are certainly orderable. In contrast, the field ofthe complex numbers C is non-orderable, because the equation x + y = 0has a non-trivial solution, x = i, y = 1, in C . The finite fields Z p and thefields of the real p-adic numbers Q p are non-orderable for similar reasons (seeRibenboim [24] p.144-145). 3 .3 Definition (Totally Ordered Rings) . Let K be an orderable ring (field)and let K + ⊂ K be a set that satisfies the properties given above. Then:1. We define the relation < K + on K by x < K + y if y − x ∈ K + . We shalloften write simply < instead of < K + if the choice of K + is clear from thecontext. Then ( K , + , · , < ), denoted for short by K , is called a totallyordered ring (field) or simply a ordered ring (field) for short. If x ∈ K ,we define the absolute value of x by | x | =: max( − x, x ).2. If A ⊂ K , we define the set of upper bounds of A by U B ( A ) =: { x ∈ K : ( ∀ a ∈ A )( a ≤ x ) } . We denote by sup K ( A ) or, simply by sup( A ),the least upper bound of A (if it exists).3. The cofinality cof( K ) of K is the cardinality of the smallest unboundedsubset of K .4. Let I be a directed set (Kelley [13], Chapter 2)). A net f : I → K iscalled fundamental or Cauchy if for every ε ∈ K + there exists h ∈ I such that | f ( i ) − f ( j ) | < ε for all i, j ∈ I such that i, j < h .5. Let a, b ∈ K and a ≤ b . We let [ a, b ] =: { x ∈ K : a ≤ x ≤ b } and( a, b ) =: { x ∈ K : a < x < b } . A totally ordered ring (field) K will bealways supplied with the order topology – with the open intervals asbasic open sets.6. A totally ordered ring (field) K is called Archimedean if for every x ∈ K ,there exists n ∈ N such that | x | ≤ n . If K is Archimedean, we also mayrefer to K ( i ) as Archimedean . (Rationals and Irrationals) . Let K be a totally ordered field.Then: (i) K contains a copy of the field of the rational numbers Q under theorder field embedding σ : Q → K defined by: σ (0) =: 0 , σ ( n ) =: n · , σ ( − n ) =: − σ ( n ) and σ ( mk ) =: σ ( m ) σ ( k ) for n ∈ N and m, k ∈ Z . We shallsimply write Q ⊆ K for short. (ii) If K \ Q is non-empty, then K \ Q is dense in K in the sense thatfor every a, b ∈ K , such that a < b , there exists x ∈ K \ Q such that a < x < b . iii) If K is Archimedean, then Q is also dense in K in the sense that forevery a, b ∈ K such that a < b there exists q ∈ Q such that a < q < b . The embedding Q ⊆ K is important for the next definition. (Infinitesimals, etc.) . Let K be a totally ordered field. Wedefine: I ( K ) =: { x ∈ K : | x | < /n for all n ∈ N } , F ( K ) =: { x ∈ K : | x | ≤ n for some n ∈ N } , L ( K ) =: { x ∈ K : n < | x | for all n ∈ N } . The elements in I ( K ) , F ( K ) , and L ( K ) are referred to as infinitesimal(infinitely small), finite and infinitely large , respectively. We sometimes write x ≈ x ∈ I ( K ) and x ≈ y if x − y ≈
0, in which case we say that x is infinitesimally close to y . If S ⊆ K , we define the monad of S in K by µ ( S ) = { s + dx : s ∈ S, dx ∈ I ( K ) } . The next result follows directly from the above definition.
Let K be a totally ordered ring. Then: (a) I ( K ) ⊂ F ( K ) ; (b) K = F ( K ) ∪ L ( K ) ; (c) F ( K ) ∩ L ( K ) = ∅ . If K is a field, then: (d) x ∈ I ( K ) if and only if x ∈ L ( K ) for every non-zero x ∈ K . Let K be a totally ordered field. Then F ( K ) is an Archimedeanring and I ( K ) is a maximal ideal of F ( K ) . Moreover, I ( K ) is a convex ideal in the sense that a ∈ F ( K ) and | a | ≤ b ∈ I ( K ) implies a ∈ I ( K ) . Conse-quently F ( K ) / I ( K ) is a totally ordered Archimedean field. Here is a characterization of the Archimedean property “in terms of in-finitesimals.” (Archimedean Property) . Let K be a totally ordered ring.Then the following are equivalent: (i) K is Archimedean. (ii) F ( K ) = K .(iii) L ( K ) = ∅ . If K is a field, then each of the above is also equivalent to I ( K ) = { } . Notice that Archimedean rings (which are not fields) might have non-zeroinfinitesimals. Indeed, if K is a non-Archimedean field, then F ( K ) is alwaysan Archimedean ring, but it has non-zero infinitesimals (see Example 2.10below). 5 .9 Definition (Ordered Valuation Fields) . Let K be a totally ordered field.Then: The mapping v : K → R ∪ {∞} is called a non-Archimedean Krullvaluation on K if, for every x, y ∈ K the properties: (a) v ( x ) = ∞ if and only if x = 0, (b) v ( xy ) = v ( x ) + v ( y ) ( Logarithmic property ), (c) v ( x + y ) ≥ min { v ( x ) , v ( y ) } ( Non-Archimedean property ), (d) | x | < | y | implies v ( x ) ≥ v ( y ) ( Convexity property ),hold. The structure ( K , v ), denoted as K for short, is called an orderedvaluation field . We define the valuation norm || · || v : K → R by the formula || x || v = e − v ( x ) with the understanding that e −∞ = 0. Also, the formula d v ( x, y ) = || x − y || v defines the valuation metric formula d v : K × K → R . We denoteby ( K , d v ) the associated metric space . (Field of Rational Functions) . Let K be an ordered field(Archimedean or not) and K [ t ] denote for the ring of polynomials in onevariable with coefficients in K . We supply the field K ( t ) =: { P ( t ) /Q ( t ) : P, Q ∈ K [ t ] and Q } , of rational functions with ordering by: f < g in K ( t ) if there exists n ∈ N such that g ( t ) − f ( t ) > K for all t ∈ (0 , /n ). Notice that K ( t ) is anon-Archimedean field: t, t , t + t , etc. are positive infinitesimals, 1 + t, t , t + t , etc. are finite, but non-infinitesimal, and 1 /t, /t , / ( t + t ),etc. are infinitely large elements of K ( t ). Also, K ( t ) is a valuation field withvaluation group Z and valuation v : K ( t ) → Z ∪ {∞} , defined as follows: If P ∈ K [ t ] is a non-zero polynomial, then v ( P ) is the lowest power of t in P and if Q is another non-zero polynomial, then v ( P/Q ) = v ( P ) − v ( Q ). We provide a collection of definitions of several different forms of complete-ness of a totally ordered field – not necessarily Archimedean. The relationsbetween these different forms of completeness will be discussed in the nexttwo sections. 6 .1 Definition (Completeness of a Totally Ordered Field) . Let K be a totallyordered field. If κ is an uncountable cardinal, then K is called Cantor κ -complete ifevery family { [ a γ , b γ ] } γ ∈ Γ of fewer than κ closed bounded intervals in K with the finite intersection property (F.I.P.) has a non-empty intersection, T γ ∈ Γ [ a γ , b γ ] = ∅ . Let ∗ K be a non-standard extension of K . Let F ( ∗ K ) and I ( ∗ K ) denotethe sets of finite and infinitesimal elements in ∗ K , respectively (see Defi-nition 2.5). Then we say that K is Leibniz complete if every x ∈ F ( ∗ K )can be presented uniquely in the form x = r + dx for some r ∈ K andsome dx ∈ I ( ∗ K ). For the concept of non-standard extension of a field we refer the reader to many of the texts on non-standard analysis, e.g.Davis [7] or Lindstrøm [18]. A very short definition of ∗ K appears also inSection 6, Example 5, of this article. K is Heine-Borel complete if a subset A ⊆ K is compact if and only if A is closed and bounded. We say that K is monotone complete if every bounded strictly increasingsequence is convergent. We say that K is Cantor complete if every nested sequence of boundedclosed intervals in K has a non-empty intersection (that is to say that K is Cantor ℵ -complete, where ℵ is the successor of ℵ = card( N )). We say that K is Weierstrass complete if every bounded sequence has aconvergent subsequence. We say that K is Bolzano complete if every bounded infinite set has acluster point. K is Cauchy complete if every fundamental I -net in K is convergent,where I is an index set with card( I ) = cof( K ). We say that K is sim-ply sequentially complete if every fundamental (Cauchy) sequence in K converges (regardless of whether or not cof( K ) = ℵ ; see Definition 2.3). K is Dedekind complete (or order complete ) if every non-empty subset of K that is bounded from above has a supremum.7 Let K be Archimedean. Then K is Hilbert complete if K is a maximalArchimedean field in the sense that K has no proper totally orderedArchimedean field extension. If κ is an infinite cardinal, K is called algebraically κ -saturated if everyfamily { ( a γ , b γ ) } γ ∈ Γ of fewer than κ open intervals in K with the F.I.P.has a non-empty intersection, T γ ∈ Γ ( a γ , b γ ) = ∅ . If K is algebraically ℵ -saturated – i.e. every nested sequence of open intervals has a non-emptyintersection – then we simply say that K is algebraically saturated . A metric space is called spherically complete if every nested sequence ofclosed balls has nonempty intersection. In particular, an ordered valu-ation field ( K , v ) is spherically complete if the associated metric space( K , d v ) is spherically complete (Definition 2.9). (Terminology) . Here are some remarks about the above termi-nology: • Leibniz completeness , listed as number 2 in Definition 3.1 above, ap-pears in the early Leibniz-Euler Infinitesimal Calculus as the statementthat “every finite number is infinitesimally close to a unique usualquantity.” Here the “usual quantities” are what we now refer to asthe real numbers and K in the definition above should be identifiedwith the set of the reals R . We will sometimes express the Leibnizcompleteness as F ( ∗ K ) = µ ( K ) (Definition 2.5) which is equivalent to F ( ∗ K ) / I ( ∗ K ) = K (Theorem 2.7). • Cantor κ -completeness , monotone completeness, Weierstrass complete-ness, Bolzano completeness and Heine-Borel completeness typically ap-pear in real analysis as “theorems” or “important principles” ratherthan as forms of completeness; however, in non-standard analysis, Can-tor κ -completeness takes a much more important role along with theconcept of algebraic saturation. • Cauchy completeness , listed as number 7 above, is equivalent to theproperty: K does not have a proper ordered field extension L such that K is dense in L . The Cauchy completeness is commonly known as se-quential completeness in the particular case of Archimedean fields (andmetric spaces), where I = N . It has also been used in constructions ofthe real numbers: Cantor’s construction using fundamental (Cauchy)8equences (see Hewitt & Stromberg [8] and O’Connor [21] and alsoBorovik & Katz [3]). • Dedekind completeness , listed as number 8 above, was introduced byDedekind (independently from many others, see O’Connor [21]) at theend of the 19 th century. From the point of view of modern mathematics,Dedekind proved the consistency of the axioms of the real numbersby constructing his field of Dedekind cuts, which is an example of aDedekind complete totally ordered field. • Hilbert completeness , listed as number 9 above, was originally intro-duced by Hilbert in 1900 with his axiomatic definition of the real num-bers (see Hilbert [9] and O’Connor [21]).To the end of this section we present some commonly known facts aboutthe Dedekind completeness (without or with very short proofs). (Existence of Dedekind Fields) . There exists a Dedekind com-plete field.Proof.
For the classical constructions of such fields due to Dedekind andCantor, we refer the reader to Rudin [26] and Hewitt & Stromberg [8], re-spectively. For a more recent proof of the existence of a Dedekind completefield (involving the axiom of choice) we refer to Banaschewski [2] and for anon-standard proof of the same result we refer to Hall & Todorov [10]. (Embedding) . Let A be an Archimedean field and R be aDedekind complete field. For every α ∈ A we let C α =: { q ∈ Q : q < α } .Then for every α, β ∈ A we have: (i) sup R ( C α + β ) = sup R ( C α ) + sup R ( C β ) .;(ii) sup R ( C αβ ) = sup R ( C α ) sup R ( C β ) ; (iii) α ≤ β implies C α ⊆ C β . Conse-quently, the mapping σ : A → R , given by σ ( α ) =: sup R ( C α ) , is an orderfield embedding of A into R . All Dedekind complete fields are mutually order-isomorphicand they have the same cardinality, which is usually denoted by c . Conse-quently, every Archimedean field has cardinality at most c . Every Dedekind complete totally ordered field is Archimedean.Proof.
Let R be such a field and suppose, to the contrary, that R is non-Archimedean. Then L ( R ) = ∅ by Theorem 2.8. Thus N ⊂ R is boundedfrom above by | λ | for any λ ∈ L ( R ) so that α = sup R ( N ) ∈ K exists. Thenthere exists n ∈ N such that α − < n implying α < n +1, a contradiction.9 Completeness of an Archimedean Field
We show that in the particular case of an Archimedean field, the differentforms of completeness (1)-(10) in Definition 3.1 are equivalent. In the caseof a non-Archimedean field, the equivalence of these different forms of com-pleteness fails to hold – we shall discuss this in the next section. (Completeness of an Archimedean Field) . Let K be a totallyordered Archimedean field. Then the following are equivalent. (i) K is Cantor κ -complete for any infinite cardinal κ . (ii) K is Leibniz complete. (iii) K is Heine-Borel complete. (iv) K is monotone complete. (v) K is Cantor complete (i.e. Cantor ℵ -complete, not for all cardinals). (vi) K is Weierstrass complete. (vii) K is Bolzano complete. (viii) K is Cauchy complete. (ix) K is Dedekind complete. (x) K is Hilbert complete.Proof. ( i ) ⇒ ( ii ) : Let κ be the successor of card( K ). Let x ∈ F ( ∗ K ) and S =: { [ a, b ] : a, b ∈ K and a ≤ x ≤ b in ∗ K } . Clearly S satisfies the finiteintersection property and card( S ) = card( K × K ) = card( K ) < κ ; thus,by assumption, there exists r ∈ T [ a,b ] ∈ S [ a, b ]. To show x − r ∈ I ( ∗ K ),suppose (to the contrary) that n < | x − r | for some n ∈ N . Then either x < r − n or r + n < x . Thus (after letting r − n = b or r + n = a ) weconclude that either r ≤ r − n , or r + n ≤ r , a contradiction.10 ii ) ⇒ ( iii ) : Our assumption (ii) justifies the following definitions: We definest : F ( ∗ K ) → K by st( x ) = r for x = r + dx , dx ∈ I ( ∗ K ). Also, if S ⊂ K , we let st[ ∗ S ] = { st( x ) : x ∈ ∗ S ∩F ( ∗ K ) } . If S is compact, then S is bounded and closed since K is a Hausdorff space as an ordered field.Conversely, if S is bounded and closed, it follows that ∗ S ⊂ F ( ∗ K )(Davis [7], p. 89) and st[ ∗ S ] = S (Davis [7], p. 77), respectively. Thus ∗ S ⊂ µ ( S ), i.e. S is compact (Davis [7], p. 78).( iii ) ⇒ ( iv ) : Let ( x n ) be a bounded from above strictly increasing sequencein K and let A = { x n } denote the range of the sequence. Clearly A \ A is either empty or contains a single element which is the limit of ( a n );hence it suffices to show that A = A . To this end, suppose, to thecontrary, that A = A . Then we note that A is compact by assumptionsince ( a n ) is bounded; however, if we define ( r n ) by r = 1 / x − x ), r n = min { r , . . . , r n − , / x n +1 − x n ) } , then we observe that thesequence of open intervals ( U n ), defined by U n = ( x n − r n , x n + r n ), isan open cover of A that has no finite subcover. Indeed, ( U n ) is pairwisedisjoint so that every finite subcover contains only a finite number ofterms of the sequence. The latter contradicts the compactness of A .( iv ) ⇒ ( v ) : Suppose that { [ a i , b i ] } i ∈ N satisfies the finite intersection prop-erty. Let Γ n =: ∩ ni =1 [ a i , b i ] and observe that Γ n = [ α n , β n ] where α n =: max i ≤ n a i and β n =: min i ≤ n b i . Then { α n } n ∈ N and {− β n } n ∈ N are bounded increasing sequences; thus α =: lim n →∞ α n and − β =:lim n →∞ − β n exist by assumption. If β < α , then for some n we wouldhave β n < α n , a contradiction; hence, α ≤ β . Therefore ∩ ∞ i =1 [ a i , b i ] =[ α, β ] = ∅ .( v ) ⇒ ( vi ) : This is the familiar
Bolzano-Weierstrass Theorem (Bartle & Sher-bert [1], p. 79).( vi ) ⇒ ( vii ) : Let A ⊂ K be a bounded infinite set. By the Axiom of Choice, A has a denumerable subset – that is, there exists an injection { x n } : N → A . As A is bounded, { x n } has a subsequence { x n k } that convergesto a point x ∈ K by assumption. Then x must be a cluster point of A because the sequence { x n k } is injective, and thus not eventuallyconstant.( vii ) ⇒ ( viii ) : For the index set we can assume that I = N since cofinality ofany Archimedean set is ℵ = card( N ). Let { x n } be a Cauchy sequence11n K . Then range( { x n } ) is a bounded set. If range( { x n } ) is finite,then { x n } is eventually constant (and thus convergent). Otherwise,range( { x n } ) has a cluster point L by assumption. To show that { x n } → L , let ǫ ∈ K + and N ∈ N be such that n, m ≥ N implies that | x n − x m | < ǫ . Observe that the set { n ∈ N : | x n − L | < ǫ } is infinite because L is a cluster point, so that A =: { n ∈ N : | x n − L | < ǫ , n ≥ N } isnon-empty. Let M =: min A . Then, for n ≥ M , we have | x n − L | ≤| x n − x M | + | x M − L | < ǫ , as required.( viii ) ⇒ ( ix ) : This proof can be found in (Hewitt & Stromberg [8], p. 44).( ix ) ⇒ ( x ) : Let A be a totally ordered Archimedean field extension of K .We have to show that A = K . Recall that Q is dense in A as it isArchimedean; hence, the set { q ∈ Q : q < a } is non-empty and boundedabove in K for all a ∈ A . Consequently, the mapping σ : A → K , where σ ( a ) =: sup K { q ∈ Q : q < a } , is well-defined by our assumption. Notethat σ fixes K . To show that A = K we will show that σ is just theidentity map. Suppose (to the contrary) that A = K and let a ∈ A \ K .Then σ ( a ) = a so that either σ ( a ) > a or σ ( a ) < a . If it is the former,then there exists p ∈ Q such that a < p < σ ( a ), contradicting the factthat σ ( a ) is the least upper bound for { q ∈ Q : q < a } and if it is thelatter then there exists p ∈ Q such that σ ( a ) < p < a , contradictingthe fact that σ ( a ) is an upper bound for { q ∈ Q : q < a } .( x ) ⇒ ( i ) : Let D be a Dedekind complete field (such a field exists by Theo-rem 3.3). We can assume that K is an ordered subfield of D by Theo-rem 3.4. Thus we have K = D by assumption, since D is Archimedean.Now, suppose (to the contrary) that there is an infinite cardinal κ and afamily [ a i , b i ] i ∈ I of fewer than κ closed bounded intervals in K with thefinite intersection property such that T i ∈ I [ a i , b i ] = ∅ . Because [ a i , b i ]satisfies the finite intersection property, the set A =: { a i : i ∈ I } isbounded from above and non-empty so that c =: sup( A ) exists in D .Thus a i ≤ c ≤ b i for all i ∈ I so that c ∈ D \ K . Thus D is a properfield extension of K , a contradiction. It should be noted that the equivalence of ( ii ) and ( ix ) aboveis proved in Keisler ([12], pp. 17-18) with somewhat different arguments.12lso, the equivalence of ( ix ) and ( x ) is proved in Banaschewski [2] using adifferent method than ours (with the help of the axiom of choice). In this section, we discuss how removing the assumption that K is Archimedeanaffects our result from the previous section. In particular, several of the formsof completeness listed in Definition 3.1 no longer hold, and those that do areno longer equivalent. Let K be an ordered field satisfying any of the following: (i) Bolzano complete. (ii)
Weierstrass complete. (iii)
Monotone complete. (iv)
Dedekind complete (v)
Cantor κ -complete for κ > card( K ) . (vi) Leibniz complete (in the sense that every finite number can be decom-posed uniquely into the sum of an element of K and an infinitesimal).Then K is Archimedean. Consequently, if K is non-Archimedean, then eachof (i)-(vi) is false.Proof. We will only prove the case for Leibniz completeness and leave therest to the reader.Suppose, to the contrary, that K is non-Archimedean. Then there existsa dx ∈ I ( K ) such that dx = 0 by Theorem 2.8. Now take α ∈ F ( ∗ K )arbitrarily. By assumption there exists unique k ∈ K and dα ∈ I ( ∗ K ) suchthat α = k + dα . However, we know that dx ∈ I ( ∗ K ) as well because K ⊂ ∗ K and the ordering in ∗ K extends that of K . Thus ( k + dx )+( dα − dx ) = k + dα = α where k + dx ∈ K and dα − dx ∈ I ( ∗ K ). This contradicts the uniquenessof k and dα . Therefore K is Archimedean.As before, κ + stands for the successor of κ , ℵ = ℵ +0 and ℵ = card( N ).13 .2 Theorem (Cardinality and Cantor Completeness) . Let K be an orderedfield. If K is non-Archimedean and Cantor κ -complete (see Definition 3.1),then κ ≤ card( K ) .Proof. This is essentially the proof of ( i ) ⇒ ( ii ) in Theorem 4.1.In the proof of the next result we borrow some arguments from a similar(unpublished) result due to Hans Vernaeve. (Cofinality and Saturation) . Let K be an ordered field and κ be an uncountable cardinal. Then the following are equivalent: (i) K is algebraically κ -saturated. (ii) K is Cantor κ -complete and cof ( K ) ≥ κ .Proof. ( i ) ⇒ ( ii ) : Let C =: { [ a γ , b γ ] } γ ∈ Γ and O =: { ( a γ , b γ ) } γ ∈ Γ be families of fewerthan κ bounded closed and open intervals, respectively, where C hasthe F.I.P.. If a k = b p for some k, p ∈ Γ, then T γ ∈ Γ [ a γ , b γ ] = { a k } by the F.I.P. in C . Otherwise, O has the F.I.P.; thus, there exists α ∈ T γ ∈ Γ ( a γ , b γ ) ⊆ T γ ∈ Γ [ a γ , b γ ] by algebraic κ -saturation. Hence K isCantor κ -complete. To show that the cofinality of K is greater than orequal to κ , let A ⊂ K be a set with card( A ) < κ . Then T a ∈ A ( a, ∞ ) = ∅ by algebraic κ -saturation.( ii ) ⇒ ( i ) : Let { ( a γ , b γ ) } γ ∈ Γ be a family of fewer than κ elements with theF.I.P.. Without loss of generality, we can assume that each intervalis bounded. As cof( K ) ≥ κ , there exists ρ ∈ U B ( { b l − a k : l, k ∈ Γ } )(that is, b l − a k ≤ ρ for all l, k ∈ Γ) which implies that ρ > ρ is a lower bound of { b l − a k : l, k ∈ Γ } . Next, we show thatthe family { [ a γ + ρ , b γ − ρ ] } γ ∈ Γ satisfies the F.I.P.. Let γ , . . . , γ n ∈ Γand ζ =: max k ≤ n { a γ k + ρ } . Then, for all m ∈ N such that m ≤ n ,we have a γ m + ρ ≤ ζ ≤ b γ m − ρ by the definition of ρ ; thus, ζ ∈ [ a γ m + ρ , b γ m − ρ ] for m ≤ n . By Cantor κ -completeness, there exists α ∈ T γ ∈ Γ [ a γ + ρ , b γ − ρ ] ⊆ T γ ∈ Γ ( a γ , b γ ). Let K be an ordered field. If K is algebraically saturated, then K is sequentially complete. roof. Let { x n } be a Cauchy sequence in K . Define { δ n } by δ n = | x n − x n +1 | .If { δ n } is not eventually constant, then there is a subsequence { δ n k } such that δ n k > k ; however, this yields T k (0 , δ n k ) = ∅ , which contradicts δ n →
0. Therefore { δ n } is eventually zero so that { x n } is eventually constant. Let K be an ordered field. If K is Cantor complete, but notalgebraically saturated, then K is sequentially complete.Proof. By Theorem 5.3, we know there exists an unbounded increasing se-quence { ǫ n } . Let { x n } be a Cauchy sequence in K . For all n ∈ N , we define S n =: [ x m n − ǫ n , x m n + ǫ n ], where m n ∈ N is the minimal element such that l, j ≥ m n implies | x l − x j | < ǫ n . Let A ⊂ N be finite and ρ =: max( A ); thenwe observe that x m ρ ∈ S k for any k ∈ A because m k ≤ m ρ ; hence { S n } satis-fies the F.I.P.. Therefore there exists L ∈ T ∞ k =1 S k by Cantor completeness.It then follows that x n → L since { ǫ n } → Let K be an ordered field, then we have the following impli-cations K is κ -saturated ⇒ K is Cantor κ -complete ⇒ K is sequentially complete . Proof.
The first implication follows from Theorem 5.3. For the second wehave two cases depending on whether K is algebraically saturated, which arehandled by Lemmas 5.4 and Lemma 5.5. In this section we present several examples of non-Archimedean fields which,on one hand, illustrate some of the results from the previous sections, buton the other prepare us for the discussion of the history of calculus in thenext section. This section alone – with the help perhaps, of Section 2 –might be described as “the shortest introduction to non-standard analysisever written” and for some readers might be an “eye opener.”If X and Y are two sets, we denote by Y X the set of all functions from X to Y . In particular, Y N stands for the set of all sequences in Y . We use thenotation P ( X ) for the power set of X . To simplify the following discussion,we shall adopt the GCH (Generalized Continuum Hypothesis) in the form2 ℵ α = ℵ α +1 for all ordinals α , where ℵ α are the cardinal numbers. Also, welet card( N ) = ℵ and card( R ) = c (= ℵ ) and we shall use c + (= ℵ ) for the15uccessor of c . Those readers who do not like cardinal numbers are advisedto ignore this remark.In what follows, K is a totally ordered field (Archimedean or not) withcardinality card( K ) = κ and cofinality cof( K ) (Definition 2.3). Let K ( t )denotes the field of the rational functions (Example 2.10). Then we have thefollowing examples of non-Archimedean fields:(1) K ⊂ K ( t ) ⊂ K ( t Z ) ⊂ K h t R i ⊂ K (( t R )) , where: The field of
Hahn series with coefficients in K and valuation group R isdefined to be the set K (( t R )) =: n S = X r ∈ R a r t r : a r ∈ K and supp( S ) is a well ordered set o , where supp( S ) = { r ∈ R : a r = 0 } . We supply K (( t R )) (denoted some-times by K ( R )) with the usual polynomial-like addition and multipli-cation and the canonical valuation ν : K (( t R )) → R ∪ {∞} defined by ν (0) =: ∞ and ν ( S ) =: min(supp( S )) for all S ∈ K (( t R )) , S = 0. As well, K (( t R )) has a natural ordering given by K (( t R )) + =: n S = X r ∈ R a r t r : a ν ( S ) > o . The field of Levi-Civita series is defined to be the set K h t R i =: n ∞ X n =0 a n t r n : a n ∈ K and ( r n ) ∈ R N o , where the sequence ( r n ) is required to be strictly increasing and un-bounded. K ( t Z ) =: n P ∞ n = m a n t n : a n ∈ K and m ∈ Z o is the field of formal Laurentseries with coefficients in K .Both K ( t Z ) and K h t R i are supplied with algebraic operations, orderingand valuation inherited from K (( t R )).16 . Since K ( t ) is Non-Archimedean (Example 2.10), so are the fields K ( t Z ), K (cid:10) t R (cid:11) and K (( t R )). Here is an example for a positive infinitesimal, P ∞ n =0 n ! t n +1 / , and a positive infinitely large number, P ∞ n = − t n +1 / = √ tt − t , both in K (cid:10) t R (cid:11) . If K is real closed, then both K h t R i and K (( t R ))are real closed (Prestel [23]). The cofinality of each of the fields K ( t Z ), K h t R i , and K (( t R )) is ℵ = card( N ), because the sequence (1 /t n ) n ∈ N isunbounded in each of them. The field K (( t R )) is spherically completeby (Krull [14] and Luxemburg [20], Theorem 2.12). Consequently, thefields K ( t Z ), K h t R i , and K (( t R )) are Cauchy (and sequentially) complete.Neither of these fields is necessarily Cantor complete or saturated. If K is Archimedean, these fields are certainly not Cantor complete. Thefact that the series R ( t Z ) and R h t R i are sequentially complete was alsoproved independently in (Laugwitz [16]). The field R h t R i was introducedby Levi-Civita in [17] and later was investigated by D. Laugwitz in [16]as a potential framework for the rigorous foundation of infinitesimal cal-culus before the advent of Robinson’s nonstandard analysis. It is also anexample of a real-closed valuation field that is sequentially complete, butnot spherically complete (Pestov [22], p. 67). Let ∗ K = K N / ∼ be a non-standard extension of K . Here K N standsfor the ring of all sequences in K and ∼ is an equivalence relation on K N defined in terms of a (fixed) free ultrafilter U on N : ( a n ) ∼ ( b n ) if { n ∈ N : a n = b n } ∈ U . We denote by (cid:10) a n (cid:11) ∈ ∗ K the equivalenceclass of the sequence ( a n ) ∈ K N . The non-standard extension of a set S ⊆ K is defined by ∗ S = (cid:8)(cid:10) a n (cid:11) ∈ ∗ K : { n ∈ N : a n ∈ S } ∈ U (cid:9) .It turns out that ∗ K is a algebraically saturated ( c -saturated) orderedfield which contains a copy of K by means of the constant sequences.The latter implies card( ∗ K ) ≥ c by Theorem 5.2 and cof( ∗ K ) ≥ c byTheorem 5.3. It can be proved that ∗ N is unbounded from above in ∗ K , i.e. T n ∈ ∗ N ( n, ∞ ) = ∅ . The latter implies card( ∗ N ) ≥ c by the c -saturationof ∗ K and thus cof( ∗ K ) ≤ card( ∗ N ). Finally, ∗ K is real closed if and onlyif K is real closed. If r ∈ K , r = 0, then (cid:10) /n (cid:11) , (cid:10) r + 1 /n (cid:11) , (cid:10) n (cid:11) presentexamples for a positive infinitesimal, finite (but non-infinitesimal) andinfinitely large elements in ∗ K , respectively. Let X ⊆ K and f : X → K be a real function. We defined its non-standard extension ∗ f : ∗ X → ∗ K by the formula ∗ f ( (cid:10) x n (cid:11) ) = (cid:10) f ( x n ) (cid:11) . It is clear that ∗ f ↾ X = f , hencewe can sometimes skip the asterisks to simplify our notation. Similarlywe define ∗ f for functions f : X → K q , where X ⊆ K p (noting that17 ( K p ) = ( ∗ K ) p ). Also if f ⊂ K p × K q is a function, then ∗ f ⊂ ∗ K p × ∗ K q is as well. For the details and missing proofs of this and other results werefer to any of the many excellent introductions to non-standard analysis,e.g. Lindstrøm [18], Davis [7], Capi´nski & Cutland [4] and Cavalcante [6]. (Free Ultrafilter) . Recall that
U ⊂ P ( N ) is a free ultrafilter on N if: (a) ∅ / ∈ U ; (b) U is closed under finitely many intersections; (c)If A ∈ U and B ⊆ N , then A ⊆ B implies B ∈ U ; (d) T A ∈U A = ∅ ;(e) For every A ⊆ N either A ∈ U , or N \ A ∈ U . Recall as well thatthe existence of free ultrafilters follows from the axiom of choice (Zorn’slemma). For more details we refer again to (Lindstrøm [18]). (Leibniz Transfer Principle) . Let K be an ordered Archimedeanfield and p, q ∈ N . Then S is the solution set of the system (2) f i ( x ) = F i ( x ) , i = 1 , , . . . , n ,g j ( x ) = G j ( x ) , j = 1 , , . . . , n ,h k ( x ) ≤ H k ( x ) , k = 1 , , . . . , n , if and only if ∗ S is the solution set of the system of equations and in-equalities (3) ∗ f i ( x ) = ∗ F i ( x ) , i = 1 , , . . . , n , ∗ g j ( x ) = ∗ G j ( x ) , j = 1 , , . . . , n , ∗ h k ( x ) ≤ ∗ H k ( x ) , k = 1 , , . . . , n . Here f i , F i , g j , G j ⊂ K p × K q and h k , H k ⊂ K p × K are functions in p -variables and n , n , n ∈ N (if n = 0 , then f i ( x ) = F i ( x ) will be missingin (2) and similarly for the rest). The quantifier “for all” is over d, p, q and all functions involved. Let ∗ R be the non-standard extension of R (the previous example for K = R ). The elements of ∗ R are known as non-standard real numbers . ∗ R is a real closed algebraically saturated ( c -saturated) field in the sensethat every nested sequence of open intervals in ∗ R has a non-empty in-tersection. Also, R is embedded as an ordered subfield of ∗ R by meansof the constant sequences. We have card( ∗ R ) = c (which means that ∗ R is fully saturated). Indeed, in addition to card( ∗ R ) ≥ c (see above),we have card( ∗ R ) ≤ card( R N ) = (2 ℵ ) ℵ = 2 ( ℵ ) = 2 ℵ = c . We also18ave cof( ∗ R ) = c . Indeed, in addition to cof( ∗ R ) ≥ c (see above) wehave cof( ∗ R ) ≤ card( ∗ R ) = c . Here are several important results ofnon-standard analysis: (Leibniz Completeness Principle) . R is Leibniz complete in the sense that every finite number x ∈ ∗ R is infinitely close to some(necessarily unique) number r ∈ R ( st : F ( ∗ R ) → R by st( x ) = r . (Leibniz Derivative) . Let X ⊆ R and r be a non-trivialadherent (cluster) point of X . Let f : X → R be a real function and L ∈ R . Then lim x → r f ( x ) = L if and only if ∗ f ( r + dx ) ≈ L for allnon-zero infinitesimals dx ∈ R such that r + dx ∈ ∗ X . In the latter casewe have lim x → r f ( x ) = st( ∗ f ( r + dx )) . The infinitesimal part of the above characterization was the
Leibniz defi-nition of derivative . Notice that the above characterization of the conceptof limit involves only one quantifier in sharp contrast with the usual ε, δ -definition of limit in the modern real analysis using three non-commutingquantifiers. On the topic of counting the quantifiers we refer to Caval-cante [6]. Let ρ be a positive infinitesimal in ∗ R (i.e. 0 < ρ < /n for all n ∈ N ).We define the sets of non-standard ρ - moderate and ρ - negligible numbersby M ρ ( ∗ R ) = { x ∈ ∗ R : | x | ≤ ρ − m for some m ∈ N } , N ρ ( ∗ R ) = { x ∈ ∗ R : | x | < ρ n for all n ∈ N } , respectively. The Robinson field of real ρ -asymptotic numbers is the fac-tor ring ρ R =: M ρ ( ∗ R ) / N ρ ( ∗ R ). We denote by b x ∈ ρ R the equivalenceclass of x ∈ M ρ ( ∗ R ). As it is not hard to show that M ρ ( ∗ R ) is a con-vex subring, and N ρ ( ∗ R ) is a maximal convex ideal; thus ρ R is an or-dered field. We observe that ρ R is not algebraically saturated, since thesequence { b ρ − n } n ∈ N is unbounded and increasing in ρ R . Consequently,cof( ρ R ) = ℵ and ρ R is Cauchy (and sequentially) complete. The field ρ R was introduced by A. Robinson in (Robinson [25]) and in (Light-stone & Robinson [19]). The proof that ρ R is real-closed and Cantorcomplete can be found in (Todorov & Vernaeve [29], Theorem 7.3 and19heorem 10.2, respectively). The field ρ R is also known as Robinson’svaluation field , because the mapping v ρ : ρ R → R ∪ {∞} defined by v ρ ( b x ) = st(log ρ ( | x | )) if b x = 0, and v ρ (0) = ∞ , is a non-Archimedean val-uation. ρ R is also spherically complete (Luxemburg [20]). We sometimesrefer to the branch of mathematics related directly or indirectly to Robin-son’s field ρ R as non-standard asymptotic analysis (see the introductionin Todorov & Vernaeve [29]). By a result due to Robinson [25] the filed R h t R i can be embedded as anordered subfield of ρ R , where the image of t is b ρ . We shall write thisembedding as an inclusion, R h t R i ⊂ ρ R . The latter implies the chain ofinclusions (embeddings):(4) R ⊂ R ( t ) ⊂ R ( t Z ) ⊂ R h t R i ⊂ ρ R . These embeddings explain the name asymptotic numbers for the elementsof ρ R . Recently it was shown that the fields ∗ R (( t R )) and ρ R are orderedfiled isomorphic (Todorov & Wolf [30] ). Since R (( t R )) ⊂ ∗ R (( t R )), thechain (4) implies two more chains: R ⊂ R ( t ) ⊂ R ( t Z ) ⊂ R h t R i ⊂ R (( t R )) ⊂ ρ R , (5) ∗ R ⊂ ∗ R ( t ) ⊂ ∗ R ( t Z ) ⊂ ∗ R h t R i ⊂ ρ R . (6) In this section we offer a short survey on the history of infinitesimal calculuswritten in a polemic-like style . The purpose is to refresh the memory ofthe readers on one hand, and to prepare them for the next section on theother, where we shall claim the main point of our article . For a more detailedexposition on the subject we refer to the recent article by Borovik & Katz [3],where the reader will find more references on the subject. • The Infinitesimal calculus was founded as a mathematical discipline byLeibniz and Newton, but the origin of infinitesimals can be traced backto Cavalieri, Pascal, Fermat, L’Hopital and even to Archimedes. Thedevelopment of calculus culminated in Euler’s mathematical inventions.Perhaps Cauchy was the last – among the great mathematicians – who20till taught calculus (in ´Ecole) and did research in terms of infinites-imals. We shall refer to this period of analysis as the
Leibniz-EulerInfinitesimal Calculus for short. • There has hardly ever been a more fruitful and exciting period in math-ematics than during the time the Leibniz-Euler infinitesimal calculuswas developed. New important results were pouring down from everyarea of science to which the new method of infinitesimals had been ap-plied – integration theory, ordinary and partial differential equations,geometry, harmonic analysis, special functions, mechanics, astronomyand physics in general. The mathematicians were highly respected inthe science community for having “in their pockets” a new powerfulmethod for analyzing everything “which is changing.” We might safelycharacterize the
Leibniz-Euler Infinitesimal Calculus as the “golden ageof mathematics.” We should note that all of the mathematical achieve-ments of infinitesimal calculus have survived up to modern times. Fur-thermore, the infinitesimal calculus has never encountered logical para-doxes – such as Russell’s paradox in set theory. • Regardless of the brilliant success and the lack of (detected) logicalparadoxes, doubts about the philosophical and mathematical legiti-macy of the foundation of infinitesimal calculus started from the verybeginning. The main reason for worry was one of the principles (ax-ioms) – now called the Leibniz principle – which claims that there existsa non-Archimedean totally ordered field with very special properties (anon-standard extension of an Archimedean field – in modern terminol-ogy). This principle is not intuitively believable, especially if comparedwith the axioms of Euclidean geometry. After all, it is much easier toimagine “points, lines and planes” around us, rather than to believethat such things like an “infinitesimal amount of wine” or “infinitesi-mal annual income” might possibly have counterparts in the real world.The mathematicians of the 17 th and 18 th centuries hardly had any ex-perience with non-Archimedean fields – even the simplest such field Q ( x ) was never seriously considered as a “field extension” (in modernterms) of Q . • Looking back with the eyes of modern mathematicians, we can now seethat the Leibniz-Euler calculus was actually quite rigorous – at leastmuch more rigorous than perceived by many modern mathematicians21oday and certainly by Weierstrass, Bolzano and Dedekind, who startedthe reformation of calculus in the second part of the 19 th century. Allaxioms (or principles) of the infinitesimal calculus were correctly chosenand eventually survived quite well the test of the modern non-standardanalysis invented by A. Robinson in the 1960’s. What was missing atthe beginning of the 19 th century to complete this theory was a proofof the consistency of its axioms; such a proof requires – from modernpoint of view – only two more things: Zorn’s lemma (or equivalently,the axiom of choice) and a construction of a complete totally orderedfield from the rationals. • Weierstrass, Bolzano and Dedekind, along with many others, startedthe reformation of calculus by expelling the infinitesimals and replac-ing them by the concept of the limit . Of course, the newly created realanalysis also requires Zorn’s lemma, or the equivalent axiom of choice,but the 19 th century mathematicians did not worry about such “minordetails,” because most of them (with the possible exception of Zermelo)perhaps did not realize that real analysis cannot possibly survive with-out the axiom of choice. The status of Zorn’s lemma and the axiom ofchoice were clarified a hundred years later by P. Cohen, K. G¨odel andothers. Dedekind however (along with many others) constructed anexample of a complete field, later called the field of Dedekind cuts , andthus proved the consistency of the axioms of the real numbers. Thiswas an important step ahead compared to the infinitesimal calculus. • The purge of the infinitesimals from calculus and from mathematicsin general however came at a very high price (paid nowadays by themodern students in real analysis): the number of quantifiers in thedefinitions and theorems in the new real analysis was increased by atleast two additional quantifiers when compared to their counterpartsin the infinitesimal calculus . For example, the definition of a limit or derivative of a function in the Leibniz-Euler infinitesimal calculusrequires only one quantifier (see Theorem 6.4). In contrast, there arethree non-commuting quantifiers in their counterparts in real analysis.In the middle of the 19 th century however, the word “infinitesimals”had become synonymous to “non-rigorous” and the mathematicianswere ready to pay about any price to get rid of them. • Starting from the beginning of the 20 th century infinitesimals were22ystematically expelled from mathematics – both from textbooks andresearch papers. The name of the whole discipline infinitesimal calculus became to sound archaic and was first modified to differential calculus ,and later to simply calculus , perhaps in an attempt to erase even theslightest remanence of the realm of infinitesimals . Even in the historicalremarks spread in the modern real analysis textbooks, the authors oftenindulge of a sort of rewriting the history by discussing the history of in-finitesimal calculus, but not even mentioning the word “infinitesimal.”A contemporary student in mathematics might very well graduate fromcollege without ever hearing anything about infinitesimals. • The differentials were also not spared from the purge – because of theirhistorical connection with infinitesimals. Eventually they were “saved”by the differential geometers under a new name: the total differentials from infinitesimal calculus were renamed in modern geometry to deriva-tives . The sociologists of science might take note: it is not unusual in politics or other more ideological fields to “save” a concept or ideaby simply “renaming it,” but in mathematics this happens very, veryrarely. The last standing reminders of the “once proud infinitesimals”in the modern mathematics are perhaps the “symbols” dx and dy inthe Leibniz notation dy/dx for derivative and in the integral R f ( x ) dx ,whose resilience turned out to be without precedent in mathematics.An innocent and confused student in a modern calculus course, how-ever, might ponder for hours over the question what the deep meaning(if any) of the word “symbol” is. • In the 1960’s, A. Robinson invented non-standard analysis and Leibniz-Euler infinitesimal calculus was completely and totally rehabilitated.The attacks against the infinitesimals finally ceased, but the straight-forward hatred toward them remains – although rarely expressed openlyanymore. (We have reason to believe that the second most hated notionin mathematics after “infinitesimals” is perhaps “asymptotic series,”but this is a story for another time.) In the minds of many, however,there still remains the lingering suspicion that non-standard analysisis a sort of “trickery of overly educated logicians” who – for lack ofanything else to do – “only muddy the otherwise crystal-clear watersof modern real analysis.” • Summarizing the above historical remarks, our overall impression is –23aid figuratively – that most modern mathematicians perhaps feel muchmore grateful to Weierstrass, Bolzano and Dedekind, than to Leibnizand Euler. And many of them perhaps would feel much happier now ifthe non-standard analysis had never been invented.
The Leibniz-Euler infinitesimal calculus was based on the existence of twototally ordered fields – let us denote them by L and ∗ L . We shall call L the Leibniz field and ∗ L its Leibniz extension . The identification of thesefields has been a question of debate up to present times. What is knownwith certainty is the following: (a) L is an Archimedean field and ∗ L is anon-Archimedean field (in the modern terminology); (b) ∗ L is a proper orderfield extension of L ; (c) L is Leibniz complete (see Axiom 1 below); (d) L and ∗ L satisfy the Leibniz Transfer Principle (Axiom 2 below). (About the Notation) . The set-notation we just used to de-scribe the infinitesimal calculus – such as L , ∗ L , as well as N , Q , R ,etc. –were never used in the 18 th century, nor for most of the 19 th century. In-stead, the elements of L were described verbally as the “usual quantities” incontrast to the elements of ∗ L which were described in terms of infinitesi-mals: dx, dy, dx , etc.. In spite of that, we shall continue to use the usualset-notation to facilitate the discussion.One of the purposes of this article is to try to convince the reader thatthe above assumptions for L and ∗ L imply that L is a complete field andthus isomorphic to the field of reals, R . That means that the Leibniz-Eulerinfinitesimal calculus was already a rigorous branch of mathematics – atleast much more rigorous than many contemporary mathematicians preferto believe. Our conclusion is that the amazing success of the infinitesimalcalculus in science was possible, we argue, not in spite of lack of rigor,but because of the high mathematical rigor already embedded in theformalism.All of this is in sharp contrast to the prevailing perception among manycontemporary mathematicians that the Leibniz-Euler infinitesimal calculuswas a non-rigorous branch of mathematics. Perhaps, this perception is due24o the wrong impression which most modern calculus textbooks create. Hereare several popular myths about the level of mathematical rigor of the in-finitesimal calculus. Myth 1.
Leibniz-Euler calculus was non-rigorous, because it was based onthe concept of non-zero infinitesimals , rather than on limits . The conceptof non-zero infinitesimal is perhaps, “appealing for the intuition,” but it iscertainly mathematically non-rigorous. “There is no such thing as a non-zeroinfinitesimal.” The infinitesimals should be expelled from mathematics onceand for all, or perhaps, left only to the applied mathematicians and physiciststo facilitate their intuition.
Fact 1.
Yes, the Archimedean fields do not contain non-zero infinitesimals.In particular, R does not have non-zero infinitesimals. But in mathemat-ics there are also non-Archimedean ordered fields and each such field con-tains infinitely many non-zero infinitesimals. The simplest example of anon-Archimedean field is, perhaps, the field R ( t ) of rational functions withreal coefficients supplied with ordering as in Example 2.10 in this article.Actually, every totally ordered field which contains a proper copy of R isnon-Archimedean (see Section 6). Blaming the non-zero infinitesimals forthe lack of rigor is nothing other than mathematical ignorance ! Myth 2.
The Leibniz-Euler infinitesimal calculus was non-rigorous becauseof the lack of the completeness of the field of “ordinary scalars” L . Perhaps L should be identified with the field of rationals Q , or the field A of the realalgebraic numbers? Those who believe that the Leibniz-Euler infinitesimalcalculus was based on a non-complete field – such as Q or A – must facea very confusing mathematical and philosophical question: How, for God’ssake, such a non-rigorous and naive framework as the field of rational numbers Q could support one of the most successful developments in the history ofmathematics and science in general? Perhaps mathematical rigor is irrelevantto the success of mathematics? Or, even worst, mathematical rigor should betreated as an “obstacle” or “barrier” in the way of the success of science. Thispoint of view is actually pretty common among applied mathematicians andtheoretical physicists. We can only hope that those who teach real analysiscourses nowadays do not advocate these values in class. Fact 2.
The answer to the question “was the Leibniz field L complete”depends on whether or not the Leibniz extension ∗ L can be viewed as a “non-standard extension” of L in the sense of the modern non-standard analysis.25hy? Because our result in Theorem 4.1 of this article shows that if theLeibniz extension ∗ L of L is in fact a non-standard extension of L , then L is a complete Archimedean field which is thus isomorphic to the field of realnumbers R . On the other hand, there is plenty of evidence that Leibniz andEuler, along with many other mathematicians, had regularly employed thefollowing principle: Axiom 1 (Leibniz Completeness Principle) . Every finite number in ∗ L is in-finitely close to some (necessarily unique) number in L ( The above property of ∗ L was treated by Leibniz and theothers as an “obvious truth.” More likely, the 18 th century mathematicianswere unable to imagine a counter-example to the above statement. Theresults of non-standard analysis produce such a counter-example: there existfinite numbers in ∗ Q which are not infinitely close to any number in Q . Myth 3.
The theory of non-standard analysis is an invention of the 20 th century and has nothing to do with the Leibniz-Euler infinitesimal calculus.We should not try to rewrite the history and project backwards the achieve-ments of modern mathematics. The proponents of this point of view alsoemphasize the following features of non-standard analysis:(a) A. Robinson’s original version of non-standard analysis was based ofthe so-called Compactness Theorem from model theory: If a set S ofsentences has the property that every finite subset of S is consistent(has a model), then the whole set S is consistent (has a model).(b) The ultrapower construction of the non-standard extension ∗ K of afield K used in Example 5, Section 6, is based on the existence of a freeultrafilter. Nowadays we prove the existence of such a filter with thehelp of Zorn’s lemma. Actually the statement that for every infiniteset I there exists a free ultrafilter on I is known in modern set theoryas the free filter axiom (an axiom which is weaker than the axiom ofchoice). Fact 3.
We completely and totally agree with both (a) and (b) above. Nei-ther the completeness theorem from model theory, nor the free filter axiom can be recognized in any shape or form in the Leibniz-Euler infinitesimalcalculus. These inventions belong to the late 19 th and the first half of 20 th century. Perhaps surprisingly for many of us, however, J. Keisler [12] invented26 simplified version of non-standard analysis – general enough to support cal-culus – which does not rely on either model theory or formal mathematicallogic. It presents the definition of ∗ R axiomatically in terms of a particu-lar extension of all functions from L to ∗ L satisfying the so-called LeibnizTransfer Principle : Axiom 2 (Leibniz Transfer Principle) . For every d ∈ N and for every set S ⊂ L d there exists a unique set ∗ S ⊂ ∗ L d such that: (a) ∗ S ∩ L d = S . (b) If f ⊂ L p × L q is a function, then ∗ f ⊂ ∗ L p × ∗ L q is also a function. (c) L satisfies Theorem 6.2 for K = L .We shall call ∗ S a non-standard extension of S borrowing the terminologyfrom Example 5, Section 6. Here are two (typical) examples for the application of theLeibniz transfer principle: The identity sin( x + y ) = sin x cos y + cos x sin y holds for all x, y ∈ L if and only if this identity holds for all x, y ∈ ∗ L (where the asterisks infront of the sine and cosine are skipped for simplicity). The identity ln( xy ) = ln x + ln y holds for all x, y ∈ L + if and only if thesame identity holds for all x, y ∈ ∗ L + . Here L + is the set of the positiveelements of L and ∗ L + is its non-standard extension (where again, theasterisks in front of the functions are skipped for simplicity).Leibniz never formulated his principle exactly in the form presented above.For one thing, the set-notation such as N , Q , R , L , etc. were not in use inthe 18 th century. The name “Leibniz Principle” is often used in the modernliterature (see Keisler [12], pp. 42 or Stroyan & Luxemburg [27], pp. 28),because Leibniz suggested that the field of the usual numbers ( L or R here)should be extended to a larger system of numbers ( ∗ L or ∗ R ), which has thesame properties, but contains infinitesimals. Both Axiom 1 and Axiom 2however, were in active use in Leibniz-Euler infinitesimal calculus. Their im-plementation does not require an ellaborate set theory or formal logic; whatis a solution of a system of equations and inequalities was perfectly clear tomathematicians long before of the times of Leibniz and Euler. Both Axiom27 and Axiom 2 are theorems in modern non-standard analysis (Keisler [12],pp. 42). However, if Axiom 1 and Axiom 2 are understood as proper axioms ,they characterize the field L uniquely (up to a field isomorphism) as a com-plete Archimedean field (thus a field isomorphic to R ). Also, these axiomscharacterize ∗ L as a non-standard extension of L . True, these two axiomsdo not determine ∗ L uniquely (up to a field isomorphism) unless we borrowfrom the modern set theory such tools as cardinality . For the rigorousness ofthe infinitesimal this does not matter. Here is an example how the formula(sin x ) ′ = cos x was derived in the infinitesimal calculus: suppose that x ∈ L .Then for every non-zero infinitesimal dx ∈ ∗ L we havesin( x + dx ) − sin xdx = sin x cos( dx ) + cos x sin( dx ) − sin xdx == sin x cos( dx ) − dx + cos x sin( dx ) dx ≈ cos x, because sin x, cos x ∈ L , cos( dx ) − dx ≈ sin( dx ) dx ≈
1. Here ≈ stands for theinfinitesimal relation on ∗ L , i.e. x ≈ y if x − y is infinitesimal (Definition 2.5).Thus (sin x ) ′ = cos x by the Leibniz definition of derivative (Theorem 6.4).For those who are interested in teaching calculus through infinitesimals werefer to the calculus textbook Keisler [11] and its more advanced companionKeisler [12] (both available on internet). On a method of teaching limitsthrough infinitesimals we refer to Todorov [28]. Myth 4.
Trigonometry and the theory of algebraic and transcendental ele-mentary functions such as sin x, e x , etc. was not rigorous in the infinitesimalcalculus. After all, the theory of analytic functions was not developed untillate in 19 th century. Fact 4.
Again, we argue that the rigor of trigonometry and the elementaryfunctions was relatively high and certainly much higher than in most of thecontemporary trigonometry and calculus textbooks. In particular, y = sin x was defined by first, defining sin − y = R x dy √ − y on [ − ,
1] and geometricallyviewed as a particular arc-lenght on a circle . Then sin x on [ − π/ , π/
2] isdefined as the inverse of sin − y . If needed, the result can be extended to L (or to R ) by periodicity. This answer leads to another question: howwas the concept of arc-lenght and the integral R ba f ( x ) dx defined in terms ofinfinitesimals before the advent of Riemann’s theory of integration? On thistopic we refer the curious reader to Cavalcante & Todorov [5].28 ames Hall is an undergraduate student in mathematics at Cal Poly,San Luis Obispo. In the coming Fall of 2011 he will be a senior and will startto apply for graduate study in mathematics. His recent senior project was on Completeness of Archimedean and non-Archimedean fields and Non-StandardAnalysis .Mathematics Department, California Polytechnic State University, SanLuis Obispo, CA 93407, USA ([email protected]).
Todor D. Todorov received his Ph.D. in Mathematical Physics fromUniversity of Sofia, Bulgaria. He currently teaches mathematics at Cal Poly,San Luis Obispo. His articles are on non-standard analysis, nonlinear the-ory of generalized functions (J. F. Colombeau’s theory), asymptotic analy-sis, compactifications of ordered topological spaces, and teaching calculus.Presently he works on an axiomatic approach to the nonlinear theory ofgeneralized functions.Mathematics Department, California Polytechnic State University, SanLuis Obispo, CA 93407, USA ([email protected]).
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