Hyperreals and Nonstandard Analysis

Following the invention of calculus, serious concerns persisted about the mathematical integrity of the method of infinitesimals. Leibniz made liberal use of infinitesimals, with great effect, but his reasoning was felt to lack rigour. The Irish bishop George Berkeley criticised the assumptions underlying calculus, and his objections were not properly addressed for several centuries. In the 1800s, Bolzano, Cauchy and Weierstrass developed the {\varepsilon}{\delta} definition of limits and continuity, which allowed derivatives and integrals to be defined without recourse to infinitesimal quantities.

In the {\varepsilon}{\delta} formalism, limits are defined by something like this: the limit of a sequence {\{x_n\}} is {x} iff, for any positive {\varepsilon}, however small, we can find a number {N} such that, for {n>N} we have {|x_n - x| < \varepsilon}. We can then define the derivative of a function in the familiar way:

\displaystyle f^\prime(x) = \lim_{\Delta x\rightarrow 0} \left(\frac{f(x+\Delta x)-f(x)}{\Delta x} \right) \,. \ \ \ \ \ (1)

The {\varepsilon}{\delta} formalism has been a source of strife for generations of students of mathematics. Is there any way to avoid it?

The Hyperreal Numbers {\mathbb{R}^*}

In the 1960s, Abraham Robinson showed that the familiar system of real numbers {\mathbb{R}} can be extended to a (much) larger set called the hyperreals. The system of hyperreal numbers, denoted {\mathbb{R}^*}, contains all the real numbers {\mathbb{R}} and also infinitesimal numbers and infinite numbers. One of the axioms used to define the hyperreals is that there exists (at least) one infinitesimal number, {\varepsilon}. The multiplicative inverse of an infinitesimal is infinite, and we usually denote {1/\varepsilon} by {\omega}, so that {\varepsilon\,\omega = 1}.

A number {\varepsilon \in \mathbb{R}^*} is infinitesimal if it is smaller than every positive real number and larger than every negative real. The only real number that is infinitesimal is {0}. An infinite number is any element of {\mathbb{R}^*} that is either greater than every real number or less than every real number.

Microscope to look at monads [from Keisler, 2000].

The hyperreals are discussed in great detail in two books by H. Jerome Keisler, an elementary text introducing calculus using hyperreals (Keisler, 2000) and a more advanced text (Keisler, 2007). Both are freely available online. We can visualise the hyperreals by imagining a cloud around each real number {x}, consisting of elements of {\mathbb{R}^*} that differ from {x} by an infinitesimal. This cloud is called the monad of x, {\mathrm{monad}(x)}, a term introduced by Leibniz. Keisler used the technique of a microscope, zooming in on the neighbourhood of {x}. The idea is shown in the Figure above.

Telescope to look at galaxies [from Keisler, 2000].

For each element of {\mathbb{R}^*}, we can consider the set of numbers that differ from {x} by a finite quantity. This set is called the galaxy of {x}, {\mathrm{galaxy}(x)}. This idea is visualised in the Figure here:

Much detail is provided in the books of Keisler. We list some important properties of {\mathbb{R}^*}:

  • The set monad(0) of infinitesimal elements is a subring of {\mathbb{R}^*}: sums, differences, and products of infinitesimals are infinitesimal.
  • Any two monads are equal or disjoint.
  • The set galaxy(0) of finite elements is a subring of {\mathbb{R}}: sums, differences, and products of finite elements are finite.
  • Any two galaxies are either equal or disjoint.
  • {x} is infinite if and only if {x^{-1}} is infinitesimal.
  • {\mathbb{R}^*} has positive and negative infinitesimals.
  • {\mathbb{R}^*} has positive and negative infinite elements.
  • There are infinitely many infinitesimals and infinitely many galaxies in {\mathbb{R}^*}.
  • The product of an infinitesimal and an infinite element may be infinitesimal, finite or infinite.

Nonstandard Analysis

For any real number {x}, the set {\mathrm{monad}(x)} contains precisely one real number, {x} itself. For any hyperreal {y \in \mathrm{monad}(x)}, we call {x} the standard part of {y}, and write {x = \mathrm{st}(y)}. The standard part function rounds off each finite hyperreal to the nearest real.

Robinson used the term nonstandard analysis for his development of calculus using hyperreal numbers. He was able to define derivatives and integrals in a direct way. The standard part may be used to define the derivative:

\displaystyle f^\prime(x) = \mathrm{st} \left(\frac{f(x+\Delta x)-f(x)}{\Delta x} \right) \,, \ \ \ \ \ (2)

where {\Delta x} is an infinitesimal. Similarly, the integral is defined as the standard part of a suitable infinite sum.

Robinson proved that the system of hyperreals is logically consistent iff the system of real numbers is consistent. This settled the centuries-old arguments about the logical soundness of arguments using infinitesimals. The classic introduction to nonstandard analysis is Robinson’s book Non-standard analysis (Robinson, 1996).

[There is much more to say. I hope to return soon to this topic.]


{\bullet} Keisler, H. Jerome, 2000: Elementary Calculus: An Infinitesimal Approach. On-line edition, revised January 2022. https://people.math.wisc.edu/~keisler/calc.html

{\bullet} Keisler, H. Jerome, 2007: Foundations of Infinitesimal Calculus. On-line Edition, https://people.math.wisc.edu/~keisler/foundations.html

{\bullet} Robinson, Abraham, 1996: Non-standard analysis, Princeton University Press, ISBN 978-0-6910-4490-3.

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