Infinitesimal

First-order properties

In extending the real numbers to include infinite and infinitesimal quantities, one typically wishes to be as conservative as possible by not changing any of their elementary properties. This guarantees that as many familiar results as possible are still available. Typically, elementary means that there is no quantification over sets, but only over elements. This limitation allows statements of the form "for any number x..." For example, the axiom that states "for any number x, x + 0 = x" would still apply. The same is true for quantification over several numbers, e.g., "for any numbers x and y, xy = yx." However, statements of the form "for any set S of numbers ..." may not carry over. Logic with this limitation on quantification is referred to as first-order logic.

The resulting extended number system cannot agree with the reals on all properties that can be expressed by quantification over sets, because the goal is to construct a non-Archimedean system, and the Archimedean principle can be expressed by quantification over sets. One can conservatively extend any theory including reals, including set theory, to include infinitesimals, just by adding a countably infinite list of axioms that assert that a number is smaller than 1/2, 1/3, 1/4, and so on. Similarly, the completeness property cannot be expected to carry over, because the reals are the unique complete ordered field up to isomorphism.

There are three categories in which a non-Archimedean number system could have first-order properties compatible with those of the reals:

1. An ordered field obeys all the usual axioms of the real number system that can be stated in first-order logic. For example, the commutativity axiom x + y = y + x holds.
2. A real closed field has all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +, ×, and ≤. This is a stronger condition than obeying the ordered-field axioms. More specifically, one includes additional first-order properties, such as the existence of a root for every odd-degree polynomial. For example, every number must have a cube root.
3. The system could have all the first-order properties of the real number system for statements involving any relations (regardless of whether those relations can be expressed using +, ×, and ≤). For example, there would have to be a sine function that is well defined for infinite inputs; the same is true for every real function.

Systems in category 1, at the weak end of the spectrum, are relatively easy to construct but do not allow a full treatment of classical analysis using infinitesimals in the spirit of Newton and Leibniz. For example, the transcendental functions are defined in terms of infinite limiting processes, and therefore there is typically no way to define them in first-order logic. Increasing the analytic strength of the system by passing to categories 2 and 3, we find that the flavor of the treatment tends to become less constructive, and it becomes more difficult to say anything concrete about the hierarchical structure of infinities and infinitesimals.

Number systems that include infinitesimals

Formal series

Laurent series

An example from category 1 above is the field of Laurent series with a finite number of negative-power terms. For example, the Laurent series consisting only of the constant term 1 is identified with the real number 1, and the series with only the linear term x is thought of as the simplest infinitesimal, from which the other infinitesimals are constructed. Dictionary ordering is used, which is equivalent to considering higher powers of x as negligible compared to lower powers. David O. Tall refers to this system as the super-reals, not to be confused with the superreal number system of Dales and Woodin. Since a Taylor series evaluated with a Laurent series as its argument is still a Laurent series, the system can be used to do calculus on transcendental functions if they are analytic. These infinitesimals have different first-order properties than the reals because, for example, the basic infinitesimal x does not have a square root.

The Levi-Civita field

The Levi-Civita field is similar to the Laurent series, but is algebraically closed. For example, the basic infinitesimal x has a square root. This field is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating-point.

Transseries

The field of transseries is larger than the Levi-Civita field. An example of a transseries is:

$${\displaystyle e^{\sqrt{\ln \ln x}}+\ln \ln x+\sum _{j=0}^{\mathrm{\infty }}{e}^x{x}^{-j},}$$

where for purposes of ordering x is considered infinite.

Surreal numbers

Conway's surreal numbers fall into category 2, except that the surreal numbers form a proper class and not a set. They are a system designed to be as rich as possible in different sizes of numbers, but not necessarily for convenience in doing analysis, in the sense that every ordered field is a subfield of the surreal numbers. There is a natural extension of the exponential function to the surreal numbers.

Hyperreals

The most widespread technique for handling infinitesimals is the hyperreals, developed by Abraham Robinson in the 1960s. They fall into category 3 above, having been designed that way so all of classical analysis can be carried over from the reals. This property of being able to carry over all relations in a natural way is known as the transfer principle, proved by Jerzy Łoś in 1955. For example, the transcendental function sin has a natural counterpart *sin that takes a hyperreal input and gives a hyperreal output, and similarly the set of natural numbers ${\displaystyle \mathbb{N}}$ has a natural counterpart ${\displaystyle {}^{\ast }\mathbb{N}}$, which contains both finite and infinite integers. A proposition such as ${\displaystyle \mathrm{\forall }n\in \mathbb{N},\sin n\pi =0}$ carries over to the hyperreals as ${\displaystyle \mathrm{\forall }n\in {}^{\ast }\mathbb{N},{}^{\ast }\sin n\pi =0}$ .

Superreals

The superreal number system of Dales and Woodin is a generalization of the hyperreals. It is different from the super-real system defined by David Tall.

Dual numbers

In linear algebra, the dual numbers extend the reals by adjoining one infinitesimal, the new element ε with the property ε² = 0 (that is, ε is nilpotent). Every dual number has the form z = a + bε with a and b being uniquely determined real numbers.

One application of dual numbers is automatic differentiation. This application can be generalized to polynomials in n variables, using the Exterior algebra of an n-dimensional vector space.

Smooth infinitesimal analysis

Synthetic differential geometry or smooth infinitesimal analysis have roots in category theory. This approach departs from the classical logic used in conventional mathematics by denying the general applicability of the law of excluded middle – i.e., not (a ≠ b) does not have to mean a = b. A nilsquare or nilpotent infinitesimal can then be defined. This is a number x where x² = 0 is true, but x = 0 need not be true at the same time. Since the background logic is intuitionistic logic, it is not immediately clear how to classify this system with regard to classes 1, 2, and 3. Intuitionistic analogues of these classes would have to be developed first.

Infinitesimal delta functions

Cauchy used an infinitesimal ${\displaystyle \alpha }$ to write down a unit impulse, infinitely tall and narrow Dirac-type delta function ${\displaystyle {\delta }_{\alpha }}$ satisfying ${\displaystyle \int F(x){\delta }_{\alpha }(x)=F(0)}$ in a number of articles in 1827, see Laugwitz (1989). Cauchy defined an infinitesimal in 1821 (Cours d'Analyse) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot's terminology.

Modern set-theoretic approaches allow one to define infinitesimals via the ultrapower construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable ultrafilter. The article by Yamashita (2007) contains bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals.

Logical properties

The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist.

In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them. A consequence of this theorem is that if there is a number system in which it is true that for any positive integer n there is a positive number x such that 0 < x < 1/n, then there exists an extension of that number system in which it is true that there exists a positive number x such that for any positive integer n we have 0 < x < 1/n. The possibility to switch "for any" and "there exists" is crucial. The first statement is true in the real numbers as given in ZFC set theory: for any positive integer n it is possible to find a real number between 1/n and zero, but this real number depends on n. Here, one chooses n initially, then one finds the corresponding x. In the second expression, the statement says that there is an x (at least one), chosen firstly, which is between 0 and 1/n for any n. In this case x is infinitesimal. This is not true in the real numbers (R) given by ZFC. Nonetheless, the theorem proves that there is a model (a number system) in which this is true. The question is: what is this model? What are its properties? Is there only one such model?

There are many ways to construct such a one-dimensional linearly ordered set of numbers, but there are two fundamentally different approaches:

1. Extend the number system so that it contains more numbers than the real numbers.
2. Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers themselves.

In 1960, Abraham Robinson provided an answer following the first approach. The extended set is called the hyperreals and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are called nonstandard.

In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal set theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system, we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number that is less, in absolute value, than any positive standard real number.

In 2006 Karel Hrbacek developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels; i.e., in the coarsest level, there are no infinitesimals nor unlimited numbers. Infinitesimals are at a finer level and there are also infinitesimals with respect to this new level and so on.

Infinitesimals in teaching

Calculus textbooks based on infinitesimals include the classic Calculus Made Easy by Silvanus P. Thompson (bearing the motto "What one fool can do another can") and the German text Mathematik fur Mittlere Technische Fachschulen der Maschinenindustrie by R. Neuendorff.

Pioneering works based on Abraham Robinson's infinitesimals include texts by Stroyan (dating from 1972) and Howard Jerome Keisler (Elementary Calculus: An Infinitesimal Approach). Students easily relate to the intuitive notion of an infinitesimal difference 1-"0.999...", where "0.999..." differs from its standard meaning as the real number 1, and is reinterpreted as an infinite terminating extended decimal that is strictly less than 1.

Another elementary calculus text that uses the theory of infinitesimals as developed by Robinson is Infinitesimal Calculus by Henle and Kleinberg, originally published in 1979. The authors introduce the language of first-order logic, and demonstrate the construction of a first order model of the hyperreal numbers. The text provides an introduction to the basics of integral and differential calculus in one dimension, including sequences and series of functions. In an Appendix, they also treat the extension of their model to the hyperhyperreals, and demonstrate some applications for the extended model.

An elementary calculus text based on smooth infinitesimal analysis is Bell, John L. (2008). A Primer of Infinitesimal Analysis, 2nd Edition. Cambridge University Press. ISBN 9780521887182.

A more recent calculus text utilizing infinitesimals is Dawson, C. Bryan (2022), Calculus Set Free: Infinitesimals to the Rescue, Oxford University Press. ISBN 9780192895608.

Functions tending to zero

In a related but somewhat different sense, which evolved from the original definition of "infinitesimal" as an infinitely small quantity, the term has also been used to refer to a function tending to zero. More precisely, Loomis and Sternberg's Advanced Calculus defines the function class of infinitesimals, ${\displaystyle \mathfrak{I}}$, as a subset of functions ${\displaystyle f:V\to W}$ between normed vector spaces by

${\displaystyle \mathfrak{I}(V,W)=\{f:V\to W|f(0)=0,(\mathrm{\forall }ϵ>0)(\mathrm{\exists }\delta >0)∍||\xi ||<\delta ⟹||f(\xi )||<ϵ\}}$,

as well as two related classes ${\displaystyle \mathfrak{O},\mathfrak{o}}$ (see Big-O notation) by

${\displaystyle \mathfrak{O}(V,W)=\{f:V\to W|f(0)=0,(\mathrm{\exists }r>0,c>0)∍||\xi ||<r⟹||f(\xi )||\le c||\xi ||\}}$, and

${\displaystyle \mathfrak{o}(V,W)=\{f:V\to W|f(0)=0,\underset{||\xi ||\to 0}{lim}||f(\xi )||/||\xi ||=0\}}$.

The set inclusions ${\displaystyle \mathfrak{o}(V,W)⊊\mathfrak{O}(V,W)⊊\mathfrak{I}(V,W)}$generally hold. That the inclusions are proper is demonstrated by the real-valued functions of a real variable ${\displaystyle f:x\mapsto |x{|}^{1/2}}$, ${\displaystyle g:x\mapsto x}$, and ${\displaystyle h:x\mapsto x^2}$:

${\displaystyle f,g,h\in \mathfrak{I}(\mathbb{R},\mathbb{R}),g,h\in \mathfrak{O}(\mathbb{R},\mathbb{R}),h\in \mathfrak{o}(\mathbb{R},\mathbb{R})}$ but ${\displaystyle f,g\notin \mathfrak{o}(\mathbb{R},\mathbb{R})}$ and ${\displaystyle f\notin \mathfrak{O}(\mathbb{R},\mathbb{R})}$.

As an application of these definitions, a mapping ${\displaystyle F:V\to W}$ between normed vector spaces is defined to be differentiable at ${\displaystyle \alpha \in V}$ if there is a ${\displaystyle T\in \mathrm{H}\mathrm{o}\mathrm{m}(V,W)}$ [i.e, a bounded linear map ${\displaystyle V\to W}$] such that

${\displaystyle [F(\alpha +\xi )-F(\alpha )]-T(\xi )\in \mathfrak{o}(V,W)}$

in a neighborhood of ${\displaystyle \alpha }$. If such a map exists, it is unique; this map is called the differential and is denoted by ${\displaystyle d{F}_{\alpha }}$, coinciding with the traditional notation for the classical (though logically flawed) notion of a differential as an infinitely small "piece" of F. This definition represents a generalization of the usual definition of differentiability for vector-valued functions of (open subsets of) Euclidean spaces.

Array of random variables

Let ${\displaystyle (\mathrm{\Omega },\mathcal{F},\mathbb{P})}$ be a probability space and let ${\displaystyle n\in \mathbb{N}}$. An array ${\displaystyle \{X_{n,k}:\mathrm{\Omega }\to \mathbb{R}\mid 1\le k\le k_n\}}$ of random variables is called infinitesimal if for every ${\displaystyle ϵ>0}$, we have: $${\displaystyle \underset{1\le k\le k_n}{max}\mathbb{P}\{\omega \in \mathrm{\Omega }\mid |X_{n,k}(\omega )|\ge ϵ\}\to 0\text{ as }n\to \mathrm{\infty }}$$ The notion of infinitesimal array is essential in some central limit theorems and it is easily seen by monotonicity of the expectation operator that any array satisfying Lindeberg's condition is infinitesimal, thus playing an important role in Lindeberg's Central Limit Theorem (a generalization of the central limit theorem).