Infinitude: the Infinite and the Infinitesimal

Prologue. Stated simply, we want to claim that corresponding to every infinitesimal on the real line, there is a transfinite stretch representing the space beyond the finite space.

Infinitude. Infinity has been always a difficult and intriguing subject for mathematicians, but they find themselves embedded in a universe which is infinite and hence forced to deal with it. Here, we want to talk about supernatural numbers which, hopefully, would make visualization of the space beyond the stars easier. We claim that corresponding to every real number, there is a supernatural number representing the infinite space.

Real Numbers. Note that one way to uniquely represent a number within the unit interval (0,1] is by an infinite binary string of the form .xxxxx... where the x's after the initial binary point represent either 0 or 1. As examples we have for the rational numbers 1, 2/3, 3/4, the unique representations .111111...., .101010...., .101111.... respectively.

Supernatural Numbers. The question we want to investigate here is that whether it is possible to give some meaning to these strings if we flip them around the binary point. Fortunately for us, computer science tell us that the appropriate meaning for ....111111. is -1 (minus one). The daring wrong argument used by the computer engineer to reach the right result is that the sequence here is the power series expansion of 1/(1-x), where x=2. Ignoring these arguments, we will accept the fact that the flipped sequences can have meaning and suggest that the flipped sequences ...xxxxxx.xxx...xxx corresponding to the real numbers xxx...xxx.xxxxx... should be called supernatural numbers.

Transfinite Stretches. Just as there are infinitesimals attached with the real numbers, we can claim that there are transfinite stretches attached with supernatural numbers. A little investigation shows that infinitesimals and transfinite stretches can be considered as duals of each other, reminiscent of the point at infinity of complex analysis. Also, it should be clear that corresponding to every transcendental number there is a supernatural number. Click here for some details. If your computer can download files fast, click here.

Epilogue. Tolerating some abuse of language, we can state that an infinitesimal is what we get when we compress and fuse a set of points of cardinality 2^\aleph_0. Similarly, we get a transfinite stretch when we keep the points of 2^\aleph_0 a finite distance apart from each other. Because of the duality between the infinitesimal and the transfinite stretch, it should be clear we need to study only one of them. In short, studying the reachable fused infinitesimal is as good as studying the unreachable transfinite stretch.