Technical PS: Infinity or Not So Much

Author’s note — I didn’t want to make Jim try to recite this material to an online audience. Wouldn’t have been fair.

Infinity versus “large beyond any hope of measurement” — what difference does it make? The difference between “guaranteed” and a qualified “maybe.”

As a specific example, how would you bet if someone suggested there’s an exact duplicate Earth existing somewhere in some parallel universe? Let’s put some numbers to it…

I’ll assume Jim’s model that contains many 93‑billion‑lightyear “observable universes,” close‑packed in a ginormous array extending out to one of two limits, either infinity or something less than infinity. For the sake of the calculation we’ll focus in on other planets with exactly as many atoms of each kind as Earth has. (From recent astronomical discoveries, there seem to be many approximately Earth-size planets even within our own galaxy.)

The question is closely related to the Infinite Monkey problem. We know Earth has a finite number of atoms. Wikipedia says the planet (mass 5.972×1024 kg) is 32% (by mass) iron, 30% oxygen, 15% silicon, 14% magnesium, plus salt and pepper to taste. With a little Chemistry fun

that works out to about 12.8×1049 atoms. That’s a huge number, sure, but it’s not infinite. (In the Monkeys problem, this would be the size of a single book.)

But if we want a planet identical to Earth, all those atoms have to be arranged the same as ours are. How many different arrangements can there be? (This is like asking in the Monkeys problem, how many possible books can there be with the same letter count?) Imagine the planet divided into slots, each one large enough to hold just one atom, and we’re going to build the planet by putting one atom in the first slot, some other atom in the second slot, and so on. That’s going to give us the factorial of 12.8×1049 possible arrangements. (Factorial of a number n means 1×2×3×4…×n.)

That factorial number is so large that my computer can’t even handle it in yyy×10zz‑style notation. In fact, it’s unphysically large and we have to correct it. For our purposes all iron atoms are identical so it doesn’t matter which iron atom went where. To rule out double-counting we have to divide the too-big number by the factorial of the iron atom count. Using the same logic, we also have to divide by the factorials of each of the other atom counts. By my arithmetic the final number of possible Earths comes to 1.54×10154. Which is huge, but it’s not infinite.

Now consider your choice of Creation process busily populating the Universe with pseudo‑Earths. If the Universe is indeed infinite, then the full roster of alternative Earth‑arrangements gets used up after the first 1.54×10154 Earth‑creations so the process creates another set, and another, and another, …, infinitely often. You, personally, are guaranteed to have an infinite number of identical copies, each of whom thinks they’re you and the only you, just like you do.

On the other hand, suppose the Universe doesn’t extend infinitely far. Maybe our observable universe with the 93 billion lightyear diameter is all there is, or maybe there’s just a few dozen locally observable globes huddled together within a Universe less than a thousand billion lightyears across. Neither case offers enough room to exhaust the 1.54×10154 possibilities and you probably are unique after all. Lucky you!

† — And Stirling’s approximation.

‡ — Actually, fewer than that because a lot of the possible arrangements, like putting all the oxygen at the center or all the iron in one hemisphere, just don’t make geochemical sense and wouldn’t happen “in the wild.”

~~ RO

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