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.
Best Answer
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.