Thetalogy → Pages → ∞ vs. ∞: Comparing Infinite Values

Lincoln Henage

December 1, 2023

∞ vs. ∞: Comparing Infinite Values

Imagine a line segment. A simple 2D shape. But this line segment has infinite points. Now imagine a ray. This shape also has infinite points, but is infinitely times as long. And a line is double that length because it extends infinitely in both directions. The question: Are there more points in a line than in a line segment?

To answer this question, we could simply cut out a section of the line that is the same size as the line segment, and see that there are still infinite remaining points in the line. Therefore, a line has more points than a line segment. There are infinite points in both figures, but one of them has more points than the other. This should make sense, because we know that infinity can be greater than infinity.

But then you can go deeper. A square or plane has infinity times as many points as a line segment or line. Then a cube has infinitely more points. This proves that there are multiple values of infinity — infinite values of infinity. You can also see this by looking at the number of natural numbers (1, 2, 3…), there are infinity of them. But there are also infinity real numbers, which there are clearly way more of.

So how do we distinguish between these different values of infinity?

In thetalogy we don’t. Infinity is a set of infinite numbers, but we don’t know the difference between these numbers. I mean, it makes sense: you can’t calculate the difference between infinite values, because they are uncountable, immeasurable! That is why we must group the infinities together, and never worry about comparing them, because we never can.