In an Infinite Universe...

...anything is possible.

Really? I don’t think so*. What might be true (but unprovable either way) is that in an infinite universe anything possible must exist somewhere – indeed an infinite number of times.

That sounds preposterous, but once you’ve looked at the mathematical analog, it seems inescapable. The question about the physical universe obviously cannot possibly be resolved (unless the universe can be proved to be not infinite); the analog is actually an interesting unresolved question in mathematics – that might yet be resolved.

The mathematical analog runs thus:

Look at the decimal expansion of any irrational number – say π, e, or √2. If you look at enough of it, there are almost equal numbers of every digit. There are roughly the same number of nines as zeroes, or as any other digit, in the first hundred digits; in the first thousand, any difference is proportionally smaller; in the first ten thousand, smaller still, and so on. The differences, as a proportion, become asymptotically smaller as the length of the expansion increases – see Digit Occurrences.

(Actually, perhaps it’s not quite any irrational number. See below.)

The same goes for pairs of decimal digits. Or triples. Or any length group you care to name.

If instead of considering all the pairs of decimal digits, you only consider pairs consisting of the first and second digits, the third and fourth digits… the (2n − 1)th and the (2n)th… then what you’re really doing is looking at the base 100 expansion. Similarly, it’s easy to see the base 1,000 expansion – it’s all the triplets of the form (3n − 2)th, (3n − 1)th, (3n)th. See Pairs & Triplets

You could look at your irrational number in other bases – binary, hexadecimal, base 7, anything you like (although doing the conversion, or doing the calculation, is non-trivial!) and you’ll find the same thing: every digit, or every string of digits, however long, will eventually occur almost exactly as often as any other string of the same length.

Consider in particular the base 128 expansion. Encode the whole of Shakespeare in 7-bit ASCII characters. That’s a long string of base 128 numbers – and occurs an infinite number of times in the base 128 expansion of π.

Actually, this is an unproven conjecture – but it’s very hard to imagine that it’s not true. If it is true, there are of course vastly greater numbers of occurrences of the whole of Shakespeare with errors – the more errors you tolerate, the more frequently you’ll find them. Of course in practice it would probably take longer than the age of the universe to find the first even vaguely recognizable version.

Now encode the world in base googolplex (or whatever it takes...) occurs an infinite number of times in the base googolplex expansion of π. At very long intervals, but they’re there.

Would an infinite universe be the same? Dunno. Is the real universe infinite anyway? If it’s not, what happens when you get to the end of it? Or is there no end, but whichever way you go, you eventually find yourself coming back where you started, like walking all the way around a sphere? (See New Scientist, November 9th, 2019, p.7: We don’t know if the universe is spherical or flat.) Even if it’s (hyper)spherical, it could still be infinite – one of the best ways of visualizing that is to look at some of M.C. Escher’s drawings.

For more about that unproven conjecture, see

* In fact I’m pretty certain about it – but since there’s no way to prove it either way I’m not sure it’s really meaningful to make the claim.

Not quite any irrational number. As far as anyone knows, only “artificial” irrational numbers break the rule. You could define a number like this, “In the decimal expansion of π, replace every other occurrence of the digit 7 with a 5,” and that number would obviously have an excess of 5s and a shortage of 7s.

But can you actually define such a number? Can you do anything with it? With an approximation to it, as precise as you like, you can obviously do all the things you can do with any other real number; but I don’t think you can do anything with its exact value. That raises a question about what “defining a number” actually means. (This question is related to “the Axiom of Choice” – I leave you to investigate that yourself...)

Can you do anything with the exact values of “normal” irrational numbers like π, √2 or e? Yes, you can – because you’ve got other, exact, ways of specifying them apart from listing an infinite string of digits.