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For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes of calculus on a sound footing.

From what I have read about hyperreal numbers I understand that they are an extension of real number system and include all real numbers and infinitesimals and infinities.

I am wondering if hyperreal numbers are used only as a justification for the use of infinitesimals in calculus or do they serve to have some other applications also (of which I am not aware of)?

Like when we extend our number system from $\mathbb{N}$ to $\mathbb{C}$ at each step there is some deficiency in the existing system which is removed in the next larger system. Thus $\mathbb{Z}$ enables subtraction which is not always possible in $\mathbb{N}$ and $\mathbb{Q}$ enables division which is not always possible in $\mathbb{Z}$. The reasons to go from $\mathbb{Q}$ to $\mathbb{R}$ are non-algebraic in nature. The next step from $\mathbb{R}$ to $\mathbb{C}$ is trivial and is based on need to enable square roots, but since the existing $\mathbb{R}$ is so powerful, the new system of complex numbers exploits this power to create rich field of complex analysis.

Does the system of hyperreal numbers use the existing power of $\mathbb{R}$ to lead to a richer theory (something like the complex analysis I mentioned earlier)? Or does it serve only as an alternative to $\epsilon, \delta$ definitions? In other words what role do the non-real hyperreal numbers play in mathematics?

Since I am novice in this subject of hyperreal numbers, I would want answers which avoid too much symbolism and technicalities and focus on the essence.

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  • $\begingroup$ Why the downvote? $\endgroup$ – Paramanand Singh Mar 7 '17 at 5:10
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Your point about successive extensions of a basic number system is very well taken. We use the successive extensions $$ \mathbb{N}\hookrightarrow\mathbb{Z}\hookrightarrow\mathbb{Q}\hookrightarrow\mathbb{R} $$ to enable easier solution of problems in algebra and geometry. The Greeks had to do everything in terms of proportions referring to natural numbers alone, and this made things cumbersome. Sticking to ordered number systems, the next extension similarly is performed not for the sake of generality but rather to facilitate applications in infinitesimal calculus. Thus, once we complete the chain of extensions to $$ \mathbb{N}\hookrightarrow\mathbb{Z}\hookrightarrow\mathbb{Q}\hookrightarrow\mathbb{R}\hookrightarrow{}^\ast\mathbb{R}, $$ we get greater facility at many levels, from elementary to research.

To give three simple examples, consider the following.

(1) The definition of the derivative becomes a finite procedure rather than an infinite limiting process. Thus the derivative of $y=f(x)$ is defined by $f'(x)=\text{st}\big(\frac{\Delta y}{\Delta x}\big)$ where $\Delta x$ is an infinitesimal $x$-increment, and "st" is the standard part function "rounding off" each finite hyperreal to the nearest real number.

(2) The definition of continuity of a function which in the A-track involves multiple alternations of quantifiers and are generally thought to be confusing to students, can be replaced by what was essentially Cauchy's definition: a function $f(x)$ is continuous if an infinitesimal $x$-increment $\alpha$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function.

(3) Terry Tao has spoken and written extensively about the expressive power of Robinson's framework for analysis with infinitesimals and its utility in research; e.g, in this 2017 publication in Discrete Analysis.

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  • $\begingroup$ If I understand correctly your answer is that hyperreals reduce the processes of analysis into finite procedures (more like algebra) and in this regard they are less confusing than the ones based on "alternation of quantifiers". I will have a look at Terry Tao's paper. And the extension to hyperreals does not add any generality but adds convenience. In terms of power it is at same level as the real number system. I will wait for other answers to get more information before I decide to accept any answer. +1 for now. $\endgroup$ – Paramanand Singh Jun 23 '16 at 15:38
  • $\begingroup$ Robinson's framework is a conservative extension of ZFC so in this sense it is "at the same level". However, this is a purely theoretical result that may be misleading as far as actual mathematical practice is concerned. This was argued specifically by Henson and Keisler in a 1986 text, and in a different way by Tao more recently. $\endgroup$ – Mikhail Katz Jun 23 '16 at 15:57
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I can think of two points of view.

The first, and IMO the most typical application, is that real analysis fails to directly support reasoning about the "sufficiently small" or "sufficiently large", and requires $\epsilon-\delta$ or other sorts of arguments to translate the reasoning into something precise.

Nonstandard analysis, however, does directly support it — e.g. for any question involving only standard numbers and functions, if there is a notion of "sufficiently small", then all of the infinitesimal numbers will be sufficiently small.

Note that this application doesn't single out a single field to be "the" hyperreals. In fact, some more sophisticated uses may require specially constructed models to have various saturation properties, or may even want to work with a hierarchy of several nonstandard models.


The second is that $\mathbb{R}^\mathbb{N}$ — the algebra of all functions from the natural numbers to the reals (i.e. sequences) — has a lot more "points" than just the natural numbers.

For an example of what I mean, consider just the subalgebra of convergent sequences. In addition to all of the natural number places (where we "evaluate" a sequence $s$ at the place $n$ by taking its $n$-th element $s_n$), there is also the infinite place $\infty$, and the value of a sequence $s$ at $\infty$ is $\lim_{n \to \infty} s_n$.

For the algebra of all sequences, there are lots of infinite places. The catch is that none of these places will be real-valued — instead, they will be hyperreal-valued.

The infinite places correspond to free ultrafilters on $\mathbb{N}$, and the field, and the value of a sequence at such a place will lie in the corresponding hyperreal field constructed as an ultrapower.

(a relevant keyword here is "stone-cech compactification")

You can do some neat things here, like give rigorous meaning to various misunderstandings of calculus; e.g. when students try to think of $\lim_{n \to \infty} a_n$ as a "thing that gets closer and closer to the limit without reaching it", one might posit what they're really thinking about is the value of the sequence $a_n$ at one of the nonstandard places.

(although, in my uninformed opinion, it's probably better to realize the vague intuition by doing nonstandard analysis in a normal way — e.g. by introducing the notion of the values $a_H$ for infinite $H$)

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  • $\begingroup$ Nice answer. I would add the point that epsilon-delta definitions are paraphrases of the shorter and more intuitive infinitesimal definitions. This is true in two senses: (1) historically the infinitesimal definitions came first, as with continuity in Cauchy 1821; and (2) even today the best way to explain the epsilon-delta definitions is to explain the concepts first using infinitesimals, and then show how paraphrasing this in terms of a parametrized notion of smallness leads to Weierstrass's definition. $\endgroup$ – Mikhail Katz Aug 3 '16 at 8:13
  • $\begingroup$ @MikhailKatz Isn't NSA non-constructive? As an example, differentiation is not a computable operator (though weirdly integration is). NSA makes it out to be "just" an expression. That ought to be a disadvantage of adopting an NSA point of view in numerical analysis and hard analysis. You can't avoid $\epsilon\delta$ in hard analysis, can you? And if you can't avoid $\epsilon\delta$, then $\epsilon\delta$ is not just a mere paraphrase. $\endgroup$ – man on laptop Jan 24 '18 at 15:18
  • $\begingroup$ @jkabrg, recent work by Sam Sanders in leading journals both in mathematics and in philosophy show that Robinson's framework for analysis with infinitesimals is actually more constructive than the Weierstrassian approach, and thereby refutes the canard that holds Robinson's framework to be less constructive... $\endgroup$ – Mikhail Katz Jan 24 '18 at 15:28
  • $\begingroup$ ...I suggest you consult his recent work that can be found for instance at MathSciNet. $\endgroup$ – Mikhail Katz Jan 24 '18 at 15:28

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