Several years ago, I went to an astronomy conference in London where Brian Cox was the main speaker. In his talk, Cox touched on the notion of the “multiverse”, reasoning that there may be an infinite number of other universes out there. What’s more, he said, if something has a non-zero probability of occurring, then it must take place somewhere in one of those universes. Everything that could possibly happen, will actually happen.

If Cox is right, it means that somewhere out there is a real universe – very similar to ours – where I was too late for his lecture and never actually got to experience it. It’s an intriguing notion that immediately got me thinking about Where’s Wally? – the children’s picture puzzle books where readers have to pinpoint Wally (known as Waldo in North America) in a crowd of similar-looking people.

It’s fun trying to track down Wally, who is unique in that he is the only person in the book wearing a red-and-white striped jumper, bobble hat and glasses. But if Cox is right, Wally doesn’t just exist; somewhere out there is an entire universe made completely of Wallys. However, the idea that there might be thousands of Wallys perturbed me, as to my mind it did not accord with common sense.

The idea that there might be thousands of Wallys perturbed me, as to my mind it did not accord with common sense

I soon forgot about my Wally worries, but they all came back to me recently when I read an article (I can’t remember by whom) that argued that if there were a finite number of particles in a particular universe, there would be only a finite number of ways to arrange them. In other words, every possible combination of particles must exist in an infinite number of universes.

I saw Wally appearing over the horizon again and this time I wasn’t going to let him lie. Casting my mind back to my university days, I remembered being told that infinity comes in two distinct types. It can be countable (i.e. discrete) where individual elements can be mapped on a one-to-one basis to the sequence of integers. Or infinity can be uncountable (i.e. continuous) where those elements cannot be mapped to integers.

One mathematical problem that was posed early on during my undergraduate degree was to prove that no matter how small a section of real numbers is taken, it is impossible to map it to the integer set. Simply put, there are far too many real numbers. Countable infinities are big, but uncountable infinities are infinitely big, which led to the inescapable conclusion that “countable” divided by “uncountable” (if we ever get round to defining it) could only ever tend to zero.

As physicists, we are still unclear if space–time is continuous or discrete, but no such problem exists in mathematics. For example, the continuous group of co-ordinates that contains our universe (three of space and one of time; other dimensions are available) will by definition have an uncountable number of continuous possible positions within it. If we think about a dartboard, there are an uncountable number of possible locations where the dart could land. And yet the dart will definitely land on one of them, which to me suggests that something with zero probability can happen.

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Of course, the converse is also true. Imagine, for example, our dartboard divided into the complete set of points represented by co-ordinates made wholly of rational numbers (countable) and also into other points represented by irrational numbers, or a mix of the two (uncountable). All points can be hit by a dart, but the mixed positions overwhelmingly dominate and must have a probability of being hit of 1.

To return to our original question: how many combinations of a finite number of particles are possible in a universe? To answer that, consider just one of them. A single particle can sit in uncountably many places along a non-zero line of finite length, which means that the arrangement of a finite number of particles in an open space must also be uncountably infinite.

Wally is very unlikely to exist in this or any other universe, even if he could in principle

So there we have it: the number of infinite universes is countable, while the number of particle combinations within them is uncountable. Wally, in other words, is very unlikely to exist in this or any other universe, even if he could in principle. Whoever originally dreamed up the phrase “Everything that can possibly happen, will actually happen”, was probably a right wally.

Finally, for all the fans of Oscar contender Everything Everywhere All at Once, it is not strictly necessary for everything to exist everywhere all at once. But then again, it might. And who knows, we may even be living in a universe where Wally turns up to collect an Oscar.