A Generalization of the Cantor-Dedekind Continuumwith Nilpotent Infinitesimals

A Generalization of the Cantor-Dedekind Continuum
with Nilpotent Infinitesimals


We introduce a generalization of the Cantor-Dedekind continuum with explicit infinitesimals. These infinitesimals are used as numbers obeying the same basic rules as the other elements of the generalized continuum, in accordance with Leibniz’s original intuition, but with an important difference: their product is null, as the Dutch theologian Bernard Nieuwentijt sustained, against Leibniz’s opinion. The starting-point is the concept of shadow, and from it we define indiscernibility (the central concept) and monad. Monads of points have a global-local nature, because in spite of being infinite-dimensional real affine spaces with the same cardinal as the whole generalized continuum, they are closed intervals with length 0. Monads and shadows (initially defined for points) are then extended to any subset of the new continuum, and their study reveals interesting results of preservation in the areas of set theory and topology. All these concepts do not depend on a definition of limit in the new continuum; yet using them we obtain the basic results of the differential calculus. Finally, we give two examples illustrating how the global-local nature of the monad of a real number can be applied to the differential treatment of certain singularities.

Abstract:2000 Mathematics Subject Classification  (primary) Keywords: Infinitesimal methods, indiscernibility, differential calculus, topology, set theory

1 Introduction

Up to 1960, when Abraham Robinson created Non-standard Analysis, actual infinitesi-mals, i.e. infinitesimals considered as numbers, in the Leibniz’s tradition , were banished from mathematical analysis by Weierstrass’ definition of limit (in the 1850s), except for a minority of mathematicians and at least one great philosopher (Charles S. Peirce). But physicists and engineers (and differential geometers such as Sophus Lie, Élie Cartan, and Hermann Weyl) refused to deprive themselves of the immense heuristic power of that notion (and rightly so!).
Today, there are two main rigorous theories of actual infinitesimals: Non-standard Analysis (NSA) , using nonexplicit invertible infinitesimals, and Smooth Infinitesimal Analysis (SIA) (F.W. Lawvere, in the late 1960s) , with nilpotent infinitesimals (i.e. infinitesimals such that , for some positive integer . But both theories are considered with suspicion by the immense majority of the mathematical community, and physicists and engineers prefer their strong intuitions.
The generalization of the usual Cantor-Dedekind continuum we propose, and the ensuing Calculus, have the following features:


– The elements of , which we call generalized real numbers, are the convergent (in the usual sense) sequences in , and those sequences that converge to are called infinitesimals (so infinitesimals are explicit). The shadow of a generalized real number is just its limit as a convergent sequence in , and from this concept we define a binary relation on that coincides with the identity of the shadows, and which we call indiscernibility (. The monad of a generalized real number () is the set of all elements of that are indiscernible from . On the set we define addition term by term, but multiplication and ordering are introduced in a different manner, using the concept of shadow. We obtain an ordered ring extension of (though it is important to take into account below); moreover, the quotient of by is an ordered field isomorphic to .
Although we can embed in (through the mapping , where is the constant sequence determined by the real number ), we must emphasize two features of that are absent from :

The product of two nonnull generalized real numbers or the square of a nonnull generalized real number may be null (if and only all the factors are infinitesimal).
Strict ordering is defined on except inside the monads (as it should be expected, since the elements of the monad of a generalized real number are indiscernible). So we have this version of the usual trichotomy property:


– We work in two modes:

The mode of potentiality, i.e. the totality of notions and concepts that can be defined within the structure .
The mode of actuality, i.e. the totality of notions and concepts that can be defined within the structure with the exception of any definition of limit.

We use the mode of potentiality emphasizing the usual definition of limit, but in the mode of actuality, in the absence of such a definition, we must introduce the fundamental concepts of generalized real number, and shadow, in the mode of potentiality. Nevertheless, we must stress that this translation is only made for the sake of definition: once defined, the two fundamental concepts are used in the mode of actuality. Every notion or concept in the mode of actuality could be translated into the mode of potentiality, but then we would renounce the intuitive and computational power of actual methods.
Our work in these two modes, sometimes simultaneously (as in the definition of differentiability), reflects our conviction that a concept of actual infinitesimal and a definition of limit are both necessary to a Calculus fit, not only for mathematicians, but also for experimental scientists.


– Each generalized real number x is indiscernible from exactly one real number: its shadow, which we denote by . In fact, each generalized real number x admits a unique decomposition as the sum of a real number (its shadow) and an infinitesimal. We denote this infinitesimal by dx, and we call it the differential of x. So we have, for each , the unique decomposition, which we call the decomposition:

For each , and , we have, as a direct consequence of the decomposition (and we stress its uniqueness!):

Although we do not use a definition of limit in , we can easily derive the basic algebraic rules of differentiation, using the decomposition.


– For each subset of , we define its monad () and shadow (), and we obtain interesting set-theoretic and topological results of preservation.
The intervals in are simply the monads of the corresponding intervals in , and the length of those that are bounded (i.e. those intervals in that are monads of bounded intervals in is the same as the length of their originals in ; for instance, the bounded open and the bounded closed intervals in are

respectively, where , and (their length is .

Intervals in do not have pointlike extremities, and this feature is reminiscent of Stoic philosophical view about segments of Space or Time ; for instance, if , and , then


– The monad of each generalized real number x has a global-local nature since it is an infinite-dimensional real affine space with the same cardinal as (more precisely, ), yet it is also a closed interval of length 0 (it is easy to prove that , so ).

We use this dual nature in two examples of differential treatment of singularities.


– For each function , where I is an open interval in , its indiscernible extensions are the functions such that

If , and is an indiscernible extension of , then is said to be differentiable at iff there exists a real number such that

with the proviso that , when such limit exists in .

(which is unique) is said to be the derivative of at , for each , and we denote it by , as usual.

So we have, when f is differentiable at :
If , then .
For each ,

This is the expression, in analytical terms, of the geometric idea associated with the concept of differentiability, according to Leibniz primeval conception:

If  is differentiable at , then the graph of  coincides locally (i.e. for infinitesimal increments of the argument around ) with its tangent at the point .

Notice that if exists in (i.e. is differentiable at , in the usual
sense)  and  is  differentiable  at ,  then  is  identical  with  this  limit;  however,

may exist in the absence of , as it is the case for and ,  defined  by (clearly,

Keeping in mind that the derivatives are always associated with indiscernible extensions, and using the definition, we obtain not only the algebraic rules of derivation, but also fundamental theorems like the Chain Rule, the Inverse Function Theorem, the Mean Value Theorem, and Taylor’s Theorem.

If exists, for each , then, among the infinity of indiscernible

extensions of , there exists exactly one that is differentiable at each ; we call

this function the natural indiscernible extension of , and we denote it by .
So is the function defined by

where denotes .

The concept of natural indiscernible extension provides a rule for the definition of the analogues (and extensions) of the usual functions of Real Analysis. For instance, the natural indiscernible extensions of exp, log, sin, cos, are the functions (where is the set of positive generalized real numbers):

defined by

We show that these functions have the same basic properties as the usual ones, and we obtain, rigorously, some identities that physicists and engineers often use intuitively. For example (since , and , as seen in III) :

2 The Generalized Real Numbers

Let () be a model of the usual real number system axioms (in any of the equivalent formulations of most calculus textbooks), and let be the set of all sequences in that are convergent for the usual absolute value in (,+,,0,1). We refer to () as the Cantor-Dedekind continuum.

Definition 2.1  Let x,y .
If is the usual limit of x in (), then we call the constant sequence (), the shadow of x, and we denote it by .
is said to be indiscernible from , and we denote it by , iff and have the same shadow.
is said to be an infinitesimal iff is indiscernible from the constant sequence (0).
The monad of , denoted by is the set of all such that is indiscernible from .
So is the set of all infinitesimals.

Clearly, the indiscernibility relation, , is an equivalence relation on , and if x is an element of , then its equivalence class for is . Indiscernibility is the first and more important binary relation defined on .

The next definition introduces a ring structure for with a kind of linear ordering.

Definition 2.2  On the set , we consider two binary operations, denoted by and , and called addition and multiplication, respectively. If and are elements of , then these operations are defined by

where at the right-hand of the previous identities we consider the obvious operations on (clearly, and .
We say that x is less than y, and we denote it by , iff , and reciprocally, we say that x is greater than y, and we denote it by , iff , where in we consider the usual linear ordering on .
The elements of and will be called positive and negative, respectively.

Proposition 2.3  a) () is a commutative ring with the constant sequences (0) and (1) as zero element and identity element, respectively.
b) The shadow mapping , defined by is an idempotent ring endomorphism, i.e.


c) is a nonnull ideal, so the sum of infinitesimals is an infinitesimal, the additive inverse of an infinitesimal is also an infinitesimal, (0) is an infinitesimal, the product of an element of and an infinitesimal is still an infinitesimal, and there is a nonnull infinitesimal.
d) The product of infinitesimals is always null, i.e.

In particular, each infinitesimal is nilpotent, since , for each .
e) An element of has a multiplicative inverse iff it is not an infinitesimal.
f) If , then

So, if we adopt the version of the usual trichotomy property expressed by the third formula above, then () may be considered an ordered ring .
g) () is archimedean, i.e.

where abbreviates , when (assuming ).
h) The mapping defined by , where ( is the usual constant sequence determined by , is a ring isomorphism of () onto (), and

So, using , we can embed () in ().

Proof  a)  Only the proofs of the associative property of multiplication and the distributive property of multiplication over addition offer some (slight) difficulty.

If , then 

b) is an immediate consequence of the usual algebraic properties of limits, and c), d) follow easily from a), b).
e) If and is not infinitesimal, then a direct calculation shows that

so, since multiplication on is associative, commutative, and is its identity element, is the multiplicative inverse of
If is infinitesimal, then we have (see a) and b)), for each :

and we conclude that is not invertible.
Finally, f), g), h) admit a quite straightforward proof.

Remark 2.4  In accordance with proposition 2.3 h), we identify with and with (), for each . For instance, we identify 0 with the infinite sequence (0) and, for each , , we identify with and with . Furthermore, from now on we shall use the symbols , , not only for the usual addition, multiplication and linear ordering on , but also for the corresponding binary operations and relation on , and we shall even drop the symbol in most formulas. For example, revisiting part of definition 2.2, we have, for each

For the additive and multiplicative powers, we simply write and instead of and (where abbreviates , when (assuming )), respectively.

In the spirit of these identifications and notational simplifications, notice that if and , then (previously denoted by coincides with the result of the scalar multiplication of the real number by the sequence x.

If and is not an infinitesimal, then we denote the multiplicative inverse of x by or ; so . We also denote (the quotient of y by x) by , as usual.

We maintain the general designation of real numbers for the elements of and call the elements of generalized real numbers.

Let us see some explicit generalized real numbers (by explicit we mean unambiguously defined as a convergent sequence of real numbers):

Example 2.5  1)  The  eventually  null  sequences  ,  , (), are nonnull infinitesimal elements of . So we can exhibit nonnull infinitesimals.
2) Let be a nonnull real number. Then:
The sequences ,, , are different elements of .

In the next proposition, which admits a simple proof, e) and f) are particularly important.

Proposition 2.6  a) .
b) (.
c) .
d) Infinitesimals are not comparable with respect to the binary relation on , i.e. if and are infinitesimals, then

e) An infinitesimal is less than any positive generalized real number and greater than any negative generalized real number, i.e. if is an infinitesimal, than

In particular:

where and are the usual sets of (strictly) positive and (strictly) negative real numbers, respectively (notice that and , by proposition 2.3 h)).
f) Each generalized real number is indiscernible from exactly one real number: its shadow, i.e.

3 The Decomposition

As a direct consequence of proposition 2.3 a), b), we have:

Proposition 3.1  If x is a generalized real number, then there is a unique infinitesimal such that

Definition 3.2  If x is a generalized real number, then we denote by dx, and we call it the differential of x.

Proposition 3.3  If x is a generalized real number then is the unique decomposition of x as the sum of a real number and an infinitesimal.

Proof. We just have to use proposition 2.3 a), c), proposition 2.6 c), proposition 3.1, and, of course, definition 3.2.

We call the decomposition stated by the previous proposition, the decomposition. Notice that the differential of a generalized real number x is already inlaid in , and since and are a constant sequence and a sequence converging to 0, in , we are entitled to express the following intuition: a generalized real number has a unique decomposition as the sum of a static part (its shadow) and a dynamic part (its differential).


Corollary 3.4  a) (.
b) (.
c) (.

The following lemma is the key to obtain the basic algebraic rules of differentiation.

Lemma 3.5  a) If , then

b) If , then

In particular, for each :

c) If and , then (with )

d) If and is not an infinitesimal, then

e) If and is not an infinitesimal, then

f) If and , then there is a unique such that

Such will be denoted by , and we have:

where and are the usual positive mth roots of and , respectively.

Proof  Only the proof of f) has some difficulty.
If , then 0 and .
So, using c) and proposition 3.3, we have:

But , since ; so

We have proven the existence (and uniqueness) of and the identity

In particular, if , then

Using c) and the result already proved (notice that , since
), we obtain:

where is the infinitesimal defined by

Then, using d),

Since the product of infinitesimals is , we have:

As an immediate consequence of the previous lemma, we obtain, using proposition 3.3, the basic algebraic rules of differentiation, without using any notion of limit in :

Proposition 3.6  a) If , then

b) If , then

In particular, for each :

c) If and , then

d) If and x is not an infinitesimal, then

e) If and x is not an infinitesimal, then

f) If and , then

We close this section with a density theorem, and a theorem relating the generalized real continuum,, to the Cantor-Dedekind continuum.

Theorem 3.7 (The Density Theorem)

a) If and y are generalized real numbers such that , then there exists such that .
b) If and are real numbers such that , then there exists such that .

Proof  a) We may choose .

b) If is an infinitesimal and , then we may choose

We already mentioned the trivial facts that is an equivalence relation on and the equivalence class of each is . On the quotient of by i.e. the set / we consider now two binary operations, denoted by and , and called addition and multiplication, respectively, and a binary relation denoted by . These operations and relation are defined by:

using, at the right-hand of the previous identities, the obvious binary operations and relation on .

It is a simple task to show that , , are well-defined, and to prove the next theorem.

Theorem 3.8  a) is an ordered field with and as zero and identity elements, respectively.
b) The mapping / , defined by , is an ordered field isomorphism of onto the Cantor-Dedekind continuum, ; so if we denote these fields simply by / and , we have:

i.e. / is isomorphic to .

As we have just seen:

If we take the monads in the structure  for points, as we do in the structure