Could "$\infty$" be understood by taking the reciprocals of the Hyperreal numbers? When learning mathematics we are told that infinity is undefined. (*)
Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this approach (something we can't do with limits). This is achieved by constructing the Hyperreal numbers $^*\mathbb{R}$, which contain the real numbers and the infinitesimal numbers (all positive numbers greater than zero, yet less than any real number).
Thinking back to (*) I wonder, could sense of infinity be made somehow by including the reciprocal infinitesimal numbers? Presumably these numbers would be defined since the infinitesimal numbers are never zero.

The symbol $\infty$ was introduced by John Wallis in the 17th century as a sign for an infinite number that can be used in calculations such as computation of areas, and he also used infinitesimals of the form $\frac{1}{\infty}$ for this purpose. The procedures relied upon by Wallis are similar to those found in modern books on the hyperreals. It may therefore be tempting to use the symbol $\infty$ as Wallis did to denote an infinite number, but this may be confusing because the symbol is often used as a part of notation such as sums of series $\sum_{i=0}^\infty u_i$ where it has no meaning other than as a reminder that a limit was taken in evaluating the series. In this sense the claim that "$\infty$ is an entity that does not exist" is a matter of convention.
