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Is set theory the right way to contemplate infinity?

Combining this result with the ideas in the third module, we can conclude that universe infinity -- all the events that ever happened or will happen anywhere in the universe -- is the same size as breathed infinity, the number of moments in one breath.

This project also shows several different sizes of infinity (discussed in the second and fifth modules). It presents a mathematically precise way of thinking about infinite sets that seems more nuanced than our intuition. And yet, perhaps the mathematician's way of thinking about infinite sets is limited. Or perhaps set theory can only capture a part of the mystery.

From the perspective of set theory, the line and the plane are the same: they are both infinite sets, and the same size. But we have many other ways to understand the line and the plane (and 3-dimensional space, and 4-dimensional space…).

A topologist will tell you that a line and plane are different for the following reason: if you take one point out of a line, you get two disconnected pieces; if you take one point out of a plane, the result is still one connected piece.

A geometer will tell you that the line is one-dimensional and the plane is two-dimensional; it takes one number to describe a point in the former, and two for a point in the latter. But, at the basic level of sets, these two mathematical objects are the same size.

Infinity is not just about being big, it's about being deep. Humans navigate space, we navigate different dimensions, different types of infinities. We also navigate the infinities of life. Inside every head and heart are infinities that we can't grasp, but can still navigate and relate with -- consciousness, memory, love, longing.

On a rough, mathematical level, all these continuous infinities are the same size. Yet, with language and thought, we have finer ways of distinguishing and understanding them. Maybe infinity isn't as "big" of a concept as it seems at first.

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You can read a short explanation of why \(|\mathbb{R}^4| = |\mathbb{R}|\), 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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