Ashkan Ek: Unintuitive Infinity

Abstract

Even if the concept of infinity is not totally out of reach of human comprehension, the size of it certainly is. Some phenomena only exist within the content of infinity, the only context in which they can make sense. As a result, these phenomena not only may sound absurd, but they might also appear equally impractical. But we will see how some of these unintuitive results that emerge from infinite sets can complement our understanding of what already is within our grasp.

Some infinities are greater than other infinities, and one-to-one correspondence between two infinite sets may or may not be possible. Is there a way to map all the points on a line segment onto all the points in a rectangle or cube? Could a curve cover a surface or volume by passing through all their points?

Some infinite sets may lead to paradoxical results. It would be impossible to duplicate a physical spherical ball that is made up of finite number of atoms; there is no way to magically have two of each atom. But what if this duplication is made possible by having the sphere be made of non-measurable infinite set?

There are also infinite sets with unexpected properties. Out of all the continuous real-valued functions on a compact interval, what portion of them are nowhere differentiable?

We will explore the concepts behind these questions, which are only a few out of many rather peculiar possibilities revealed by infinite sets.

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