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What does it mean to have different sizes of infinity? What does it mean to you?

The first thing to point out is that this section, and in fact this whole project, works with the assumption that time and space are continuous phenomena.

On the other hand, there are strong theoretical arguments coming from quantum physics that everything -- time, space, energy -- is actually quantized into discrete pieces. In this case, there is a smallest discrete chunk of time (the Planck time, around \(10^{-43}\) seconds), and everything is an accumulation of these chunks.

The implication would be that time, space, and energy are discrete phenomena, so there is a finite number of instants in one second, and only a countably infinite number of instants in eternity.

For the sake of having more fun (an essential difference between the mathematician’s worldview and the physicist’s worldview), we are ignoring quantum theory in this project.

With it, there is only one size of infinity manifested in physical reality -- countable infinity.

Without it, there are at least two.

This section explains these two sizes of infinity -- one discrete and infinitely countable, the other continuous and infinitely deep. We see that, intuitively, it is a contrast between continuous and discrete, analog and digital, depth and breadth.

What does it mean, to have different sizes of infinity? What does it mean to you? We've only shown how mathematicians answer the first question. The fascinating question is the second one. Hidden below the surface is a question about definitions: how does the mathematical notion of infinity compare with the intuitive, rigorous, or non-rigorous definitions that each of us carries?

"What does it mean, to have different sizes of infinity? What does it mean to you? We've only shown how mathematicians answer the first question. The fascinating question is the second one."

What are the implications of acknowledging different sizes of infinity?

Here we give one possible example. The modern digital era is involved in approximating lived life with digital models. The conventional wisdom is that for every problem there is, or will soon be, an app to solve that problem.

Digital technology has grown so fast, for so long, seemingly unconstrained by any physical limits, that it almost seems inevitable that digital solutions will converge onto every problem and mystery of life.

It is possible, however, that we are confusing our sizes of infinity.

As discussed in the first module, the best that digital technology can do is grow towards countable infinity.

As discussed in this module, life itself comprises of a larger and deeper size of infinity.

The idea that, given enough time, digital life will grow endlessly and catch up to lived life, may simply be a misunderstanding of how big countable infinity is compared to breathed infinity.

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You can read a short explanation of why \(|(0,1)| > |\mathbb{N}|\), or an in-depth explanation (a rigorous proof), or you can take our word for it, and go read the story or the meta-discussion.

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