4.3 The multiverse and all of its universes are finite
Now we are ready to work out the answer to the question at the beginning of the chapter: is the multiverse finite or infinite? Let us start by supposing that every universe in the multiverse contains an infinite number of events/interactions. Such a universe would have to be infinitely large or have events that go on forever in time, or both, which, as we shall see, is how most cosmologists think of our own universe: at least, they regard our universe as extending in space infinitely. In such a universe, suppose that there are n events which are all independent of each other – that means that none of the events is entangled with any of the others. We can always find any number of such independent events just by making sure that none of them is in the future light cone of any other, for example, like Alice and Bob in the figure with future light cones extending from their experiment. Each of these events must have at least two outcomes – for instance, spin-up and spin-down, or, to take another example, each of the very many possible positions for an electron being detected on the hemispherical screen. (However, we do not expect the number of distinguishable positions to be infinite, because there is an inherent blurriness at scales of the Planck length, such that it would be impossible to distinguish two adjacent points separated by less than the Planck length. It may well be, though, that we can consider the points to take up a much larger size than the Planck length: for instance, an atom on the detecting screen.)
So, in the multiverse, there must be enough parallel universes, N, to contain all of the different combinations of the outcomes of these n interactions. If there were, for instance, just three spin interactions in a universe, so that n = 3, then we would need at least 23 = eight different universes: at least one for each of (1) up-up-up; (2) up-up-down; (3) up-down-up; (4) up-down-down; (5) down-up-up; (6) down-up-down; (7) down-down-up; and (8) down-down-down. Of course, if any of the interactions were not just a simple spin-filter interaction, but an electron being detected on a screen, there would be many more. In general, though, we can see that, for n events, there must be at least 2n parallel universes containing the outcomes of these events.
Now, suppose that the number, n, of these independent events is just countably, as opposed to uncountably, infinite. Then the number of parallel universes, N, needed to contain these outcomes, namely 2n, will be uncountably infinite (since, as I said in the previous Step 4.2 (Sizes of infinity), 2n is uncountably infinite if n is countably infinite). So, in that case, if N is the number of universes in the multiverse, then N must be uncountably infinite. Of course, if the number n were uncountably infinite rather than just countably infinite, N would still be uncountably infinite. So we know that, if the universes in the multiverse contain infinite numbers of events, whether countable or uncountable, then, either way, the number of universes in the multiverse must be uncountably infinite to contain them all.
Uncountable infinity is a powerful concept – an uncountably infinite number of universes in the multiverse would mean that every universe in the multiverse is replicated an uncountably infinite number of times! Here’s why: in any universe, if we trace the cause of any event successively backwards in time, we shall ultimately arrive at the same root event that is common to all universes. (This is like tracing back through the branches of a tree back to the trunk in Step 4.1 (A toy multiverse)). If there is an uncountable infinity of universes in the multiverse (each containing the root event), then the number of universes containing any of the immediate outcomes of the root event must also be uncountably infinite. In turn, the number of universes containing any of the outcomes of events caused by these outcomes will also be uncountably infinite. Ultimately, we will find that every outcome of every event in an uncountably infinite multiverse must be contained in an uncountably infinite number of universes.
From Step 3.1 (Patterns in an indeterministic universe)
The algebra of quantum mechanics says, if Alice performs a sequence of 100 of her 60º experiments, then there will be close to 75 results of spin-up, but it cannot predict exactly how many such results there will be in any given sequence. The histogram shows the expected distribution of results.
So, let’s see what happens in such a multiverse if Alice repeats her 60º experiment 100 times in succession, just as she did for the histogram in Step 3.1 (Patterns in an indeterministic universe). Inspecting the histogram, it was obvious that any given version of Alice was most likely to be in a universe where, out of the 100 experiments, she found 75, or close to 75, outcomes with spin-up. It was obvious because the bars of the histogram close to 75 were much taller than, say, those for fewer than 60 spin-up outcomes or more than 90 spin-up outcomes – these were so small that you couldn’t even see them on the chart. But, with infinity it’s different. In the uncountably infinite multiverse that we are considering, all of the bars are uncountably infinitely tall. You can’t say that the bar representing the uncountably infinite number of universes with 75 spin-up outcomes is taller or smaller than a bar representing any of the other uncountably infinite number of universes. So, Alice’s experiment to demonstrate the Born rule will not be successful: you can’t even define the likelihood of Alice being in a universe where she can confirm the Born rule.
(a) The Lebesgue measure gives a 10% chance of picking a real number between 0.4 and 0.5.
(b) In an uncountably infinite number of universes, the Lebesgue measure can be used to give the probability of Alice being in a universe with a given percentage of spin-up outcomes, if the only event in every universe is the experiment itself.
However, there might be a workaround. The number of real numbers between zero and one is uncountable, which means that the probability of picking a particular random number between zero and one is literally zero. Given that, you might conclude that you can’t predict the probability of quite randomly picking a real number with a value between, say, 0.4 and 0.5, because adding up all those zero probabilities still gives zero. However, your intuition tells you that the chance must be one in ten: just look at the shaded area that I have drawn along the number line. The chance that a dart thrown randomly along the number line will land in the shaded area is just that: 1/10.
In fact, this is a standard mathematical tool for dealing with infinite sets – it’s called the Lebesgue measure (after the French mathematician, Henri Lebesgue), and we can use it, for instance, if we can somehow arrange for the elements of the sets that we are interested in to be strung out along the number line, (a) in the figure, (or it could be an area rather than the number line, or a volume, or an even higher-dimensional space). So, taking the histogram bars, we start by assuming that they represent a finite number of universes, and we lay them in turn, end-to-end, along a horizontal line as shown in the figure (b). The proportions of the line shown for 71%, 73%, and so on, will not change as the number of universes increases. Indeed, we can assume that, in the limit of an uncountably infinite number of universes, where the horizontal line becomes the number line from 0 to 1, these proportions are still exactly as shown in the figure. So, if Alice’s 60º experiment is the only event in each universe, then the Lebesgue measure lets us see, for example, that, in an uncountably infinite number of universes, it is 95% likely that a version of Alice is in a universe where she finds an outcome within 8% of 75%.
However, in our proposed scenario, it is the countably infinite number of other events, n, which are present in addition to Alice’s experiments, that ultimately prevents us from using the Lebesgue measure. To see why, imagine an uncountably infinite number of universes where the only event is Alice performing just one 60º experiment, so that there are only two sets of universes: one containing an uncountably infinite number of universes where the outcome is spin-up and the other containing an uncountably infinite number of universes where the outcome is spin-down. Again, if there were no other events in these universes, the Lebesgue measure could be used. Since, according to our supposition, there is only one event in the universe, we could designate any real number between zero and 0.75 as representing a (single-event) universe where the outcome is spin-up and any real number greater than 0.75 up to 1.0 as a universe where the outcome is spin-down.
If there were just one other event in these universes, say B1, with its attendant outcomes of spin-up and spin-down, then the set representing universes with Alice’s spin-up would have to be split into two distinct sets – one representing universes where Alice’s result is spin-up and the outcome of B1 is also spin-up, and the other one representing universes where Alice’s result is still spin-up but the outcome of B1 is spin-down. Because of the two alternative outcomes of event B1, no universe in the first set is the same as any universe in the second set, and so these two sets don’t “overlap” in any way – technically, they are said to be non-intersecting. The set where Alice measures spin-down would similarly be split into two by the event B1, giving a total of four non-intersecting sets.
If we include another event, B2, then this splits each of the four sets again, and, as we go on like this, we see, with a countably infinite number, n, of independent events in each universe where Alice performs her experiment, that the total number of sets is 2n+1, which is uncountably infinite. For the same reason as above, none of these sets intersects with any other. This account was for Alice just doing one experiment, but the same applies if she performs, say, 100 experiments in a sequence, in an attempt to show consistency with the Born rule, and so we end up with an uncountably infinite number of non-intersecting sets.
This is why the Lebesgue measure fails: the measures of an uncountably infinite number of sets would have to be added together, but the mathematics of the Lebesgue measure requires that the non-intersecting sets forming the sum must be countable – the Lebesgue measure will not work with so-called “uncountable additivity”. So, despite the original hope of using the Lebesgue measure, it turns out that, in this case, the original intuition was correct: experiments to check the Born rule will not work in an uncountably infinite multiverse!
In summary, if the universes in the multiverse contain an infinite number of events, even just a countably infinite number, with the result that the number of universes in the multiverse is uncountably infinite, then, in any universe, sequences of experiments to measure quantum probabilities will fail. Since we do not live in such a universe – we can consistently show by experiment that the Born rule works – the conclusion is (1) there is not an uncountably infinite number of universes in the multiverse; and (2) the individual universes in the multiverse contain only finite numbers of events (because infinite numbers of events, even just countably infinite, would lead to an uncountably infinite number of universes in the multiverse).
We saw above that the number of universes, N, needed to accommodate n events where there are two outcomes for each event is at least 2n. If there had been three outcomes to each event, then N would be 3n, and so on. As long as the number of events, n, and the number of possible outcomes for each event are both finite, then N will be finite.
So we have two remarkable conclusions:
- The number of universes in the multiverse is finite; and
- The number of quantum events in any universe is finite.
From the second conclusion, we are living in a universe in which, stretching over all of space and all of time, there is only a finite number of interactions. Our universe does not go on forever, neither in space nor in time! Our universe is finite!
