How Humans Keep Calculating Infinity

The day I finally understood infinity, I did what I think most humans do with anything vast and mysterious: I tried to wrap my head around it. 😄 What is this thing? How big is it, really? And the question that quietly obsessed me: could you ever actually calculate it?

We are a species addicted to progress. We measure, we map, we count. So infinity feels almost like an insult: a thing that flat-out refuses to be measured.

But here is what I have slowly realized while growing up. The more we learn, the shorter the list of infinities gets. One by one, things we once swore were endless turn out to have an edge, a number, a finite answer.

"Infinity," it turns out, is often just the name we give to something we have not measured yet.

So can you calculate infinity? Yes, I think so, but only once you know which of the three types you are holding. 👇

🌫️ Type 1: The Infinities We Retired

These were never truly infinite. They were just unknown and enormous, and "infinite" was our placeholder for "too big to count." Then we built better instruments (or simply found smarter ways to calculate), and the placeholder fell away.

Notice the pattern in every single row. It is almost a recipe:

😉 The recipe humanity keeps using:
1. Assume it is infinite → 2. Build a better instrument (or just a smarter way to calculate) → 3. Measure it → 4. Repeat with the next "infinity."

🎢 "But some numbers are so big they must basically be infinite, right?" Nope. A googol is 10¹⁰⁰, already more than the number of atoms in the entire observable universe. Graham's number is so vast you could not write it even if every atom in the cosmos were a single digit. And yet… both are perfectly finite. You can always add 1. That is the secret: infinity is not "really, really big." It is a different kind of thing entirely.

♾️ Type 2: The Infinities That Are Real

Here it gets humbling. Some infinities are not ignorance at all: they are provably endless. You do not measure these with a telescope; you prove them with a pencil.

• The counting numbers (1, 2, 3, …) never stop, for the most obvious reason: whatever you call the biggest, I will just add 1. The more beautiful version is Euclid's proof (about 2,300 years ago) that the primes never run out.
• π is the poster child. (These days, 3.1415. As a kid, a lot more.) It is the ratio of a circle's circumference to its diameter, with decimals that run forever and never settle into a repeating loop. And we do not just suspect this: Johann Lambert proved π is irrational in 1761, so we know the digits never end. (High-performance servers have still cranked out 314 trillion of them, just to admire it. The part we cannot yet prove is whether those digits are perfectly "random.")

And then the fact that genuinely broke my brain:

⭐ Not all infinities are the same size. There are infinitely many counting numbers. There are also infinitely many real numbers just between 0 and 1. But in the 1800s, Georg Cantor proved the second infinity is strictly bigger than the first. There is not one infinity; there is a whole staircase of them, each larger than the last. 🤯

🏨 Want that weirdness in a single image? Picture Hilbert's Hotel: infinitely many rooms, every one full. A new guest walks in, and you still fit them by asking every guest to shift over by one room. Infinity plays by its own rules.

🕳️ Type 3: The Infinities We Have Not Cracked

The third kind is the most thrilling, because it is where we are still standing at the edge of the map. These infinities pop out of our best theories, and that is usually a sign the theory itself is incomplete.

• The heart of a black hole. Einstein's relativity says a collapsing giant star crushes its matter to a point of zero volume. Mass ÷ zero volume = infinite density. But almost no physicist believes nature actually holds an infinity here; it is a flare sent up by the math saying "you are missing a theory." We still do not have the quantum gravity that might reveal what is really down there.
• The first instant of the Big Bang. Rewind 13.8 billion years and everything collapses to a single point of infinite temperature and density: the same flare, the same missing theory.
• Is the whole universe infinite? The part we can see is finite (~93 B light-years across). But beyond our horizon? Maps from NASA's WMAP and ESA's Planck show space is flat to within less than 1%. A flat universe is consistent with stretching infinitely in every direction, though flatness alone does not prove it; space could still quietly curve back on itself. Genuinely open. 🔭
• And the future? In 1998 we found dark energy pushing the universe apart faster and faster. Gravity cannot reel it back, so as best we can tell, time just… keeps going. Forward, endlessly. (But what is time, really? That is a rabbit hole deep enough for its own post one day. ⏳)

🌌 So… Can You Calculate Infinity?

Honestly? I know we can, 100%. It is just a matter of how, and when. We have done it over and over already in Type 1: every telescope, every microscope, every equation quietly shrank the list of "endless" things. That is the optimistic story of science: each generation hands the next a shorter list of infinities.

And the deeper I go, the more I suspect the ones that remain (the real infinities in math, the theory-breaking ones in physics) are not endless because the universe is taunting us. They are endless because we probably just have not asked the right questions yet. Every singularity is really the cosmos whispering "you do not have the full theory." Even in pure math, there are infinities we have proven we cannot currently settle with the tools we have. The entire history of Type 1 was nothing but sharper questions and better instruments turning the impossible into a number, and there is no rule saying that history is finished.

Maybe that is the real lesson. Infinity is not so much a property of the universe as a marker of the edge of our knowledge. It is the frontier, and frontiers are made to be crossed. 🚀

Coming out with my next blog soon… ✍️
Have a nice time… ☀️
Contact me at [email protected] 📬