The presence of infinitesimals is traced back to some of the most general algebraic structures, namely, semigroups, and in fact, magmas, [1], in which none of the structures of linear order, field, or the Archimedean property need to be present. Such a clarification of the basic structures from where infinitesimals can in fact emerge may prove to have a special importance in Physics, as seen in [4-16]. The relevance of the deeper and simpler roots of infinitesimals, as they are given in Definitions 3.1 and 3.2, is shown by the close connection in Theorem 4.1 and Corollary 4.1 between the presence of infinitesimals and the non-Archimedean property, in the particular case of linearly ordered monoids, a case which, however, has a wide applicative interest.
the first to introduce the idea of infinitesimals, and show their usefulness in Calculus. And for the next two centuries, until in the second part of the 1800s Weierstrass introduced modern rigour into the subject, Calculus had much been based on a variety of intuitive, rather than rigorous uses of infinitesimals popping up in all kind of places and under any number of forms. In fact, in engineering or physics courses of Calculus, such loose appeal to infinitesimals has gone on until more recently.
As it happens, at various times, a number of facts have not been understood quite clearly related to the status of infinitesimals.
One such fact is that the Archimedean structure of the field R of usual real numbers does not allow the presence of infinitesimals. This is the reason why after the reform introduced by Weierstrass there has no longer been a place in Calculus for infinitesimals. In this regard, Robinson’s field * R of nonstandard reals happens to be non-Archimedean, and as such, proves to be able to accommodate infinitesimals.
However, in pursuing Nonstandard Analysis, Robinson had further goals in addition to obtaining a rigorous foundation for infinitesimals. Indeed, among such goals was that * R is a linearly ordered field extension of R. Furthermore, it was aimed that a good deal of the usual properties of R would automatically remain valid for * R as well, under the so called transfer principle.
A consequence of the above has been the tacit association of infinitesimals with :
• linear orders,
• fields,
• the Archimedean property.
As shown in this paper, however, the presence of infinitesimals can be traced back to far more general algebraic structures, namely, semigroups, and in fact, magmas, [1], in which none of the above three structures need to be present. Such a clarification of the basic structures from where infinitesimals can in fact emerge may prove to have a special importance in Physics, as seen in [4][5][6][7][8][9][10][11][12][13][14][15][16].
The relevance of the deeper and simpler roots of infinitesimals, as they are given in Definitions 3.1 and 3.2, is shown by the close connection in Theorem 4.1 and Corollary 4.1 between the presence of infinitesimals and the non-Archimedean property in the particular case of linearly ordered monoids, a case which, however, has a wide applicative interest.
It is useful to start by recalling the seldom considered and surprisingly rich and complex structure of the set of additive subgroups in * R.
First, we recall that R is an additive subgroup of * R. Further, we can distinguish the following four subgroups in * R, namely
where Mon(0) denotes the set of infinitesimals, that is, the so called monad at 0 ∈ * R, while F in(0) denotes the set of finite elements
Clearly, when seen from R, the subgroups (2.1) collapse to only two trivial instances, namely, {0} and R itself.
Now the important fact to note is that there are many more additive subgroups in * R, than listed in (2.1). Indeed, let ǫ ∈ Mon(0), ǫ > 0, that is, a positive infinitesimal. Then, associated with this infinitesimal ǫ we obtain the following infinitely many pair-wise disjoint additive subgroups in * R, namely (2.2) Rǫ, Rǫ 2 , Rǫ n , . . .
And in fact, there are uncountably many of that type of pair-wise disjoint additive subgroups associated with the given infinitesimal ǫ, namely
Similarly, if we take any X ∈ * R \ F in(0), X > 0, then associated with this infinitely large X we obtain the following infinitely many pair-wise disjoint additive subgroups in * R, namely
Needless to say, the additive subgroups in * R are far from being exhausted by those in (2.1) -(2.4).
As for the complexity of the relationships between various additive subgroups in * R, we can note the following.
Let ǫ, η ∈ Mon(0), ǫ, η > 0, then
and as is well known, the relation in the right hand of (2.5) is highly atypical among infinitesimals ǫ, η ∈ Mon(0).
where again, the relation in the right hand of (2.6) is highly atypical among infinitely large X, Y ∈ * R \ F in(0).
Let us now compare the above with the situation of additive subgroups of R.
hence no trace of the rich complexity of additive subgroups such as in (2.1) -(2.7).
Let us consider another example with infinitesimals, one that is closely connected with the reduced power algebras, and in particular, with * R, see [4][5][6][7][8][9][10][11][12][13][14][15][16], namely the algebra R N .
First we recall that we have the group isomorphism
Further, in the algebra R N one can distinguish the following additive semigroups
where I R N , A R N and B R N are, respectively, the set of sequences x = (x 0 , x 1 , x 2 , . . .) ∈ R N , which converge to 0 ∈ R, converge to some element in R, respectively, are bounded.
In (2.9), in view of [4][5][6][7][8][9][10][11][12][13][14][15][16], one can see I R N as the monad of 0 ∈ R N , that is, the set of infinitesimals in R N , while R N \ B R N can be seen as the set of infinitely large elements
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