This is called a geometric series. At first sight, you might guess that adding an infinite series of numbers together one after the other, forever, should lead to an infinitely large number, but this isn’t necessarily the case. For this particular series, the result of adding them all up is 1. You can see this by using a little simple algebra.
Let’s call the sum of this series S:
Now consider a different series; the original one, but with each term divided by 2:
Now subtract S/2 from S. Every term in the series disappears except the first term in S:
Adding an infinite series of numbers together is something that we can do, at least in this example, and get a finite answer.
The type of infinity we’re thinking about here is an infinite set of fractions; ½, ¼, ⅛, and so on. How many of these fractions are there? An infinite number, and we assumed this in our proof because for every fraction in the infinite series S, other than ½, there was a corresponding fraction in the infinite series S/2 to cancel it out. But this raises an interesting question. We subtracted an infinite number of fractions from an infinite number of fractions and we had one term left: ½. Does this mean that there was one more fraction in S than in S/2? The answer is no; the two infinites are precisely the same. The first mathematician to think about what we mean when we speak of an infinite set of numbers was the German mathematician Georg Cantor, at the end of the nineteenth century.
Consider, for example, the set of all integers; 1,2,3,4 … We could imagine making a table of the integers by writing them all down in a vertical column from 1 to infinity. We could then write each of the terms in our set S alongside in a neighbouring vertical column. Each fraction – ½, ¼, ⅛ – would be paired up with an integer, all the way down the list. We could do the same for our set S/2; the column would begin with ¼ rather than ½, but it would carry on all the way to infinity. This one-to-one correspondence between all three sets of numbers is the reason why Cantor claimed that the three sets have the same ‘infinity’ of numbers contained within them. Mathematicians would say that the sets have the same cardinality.
There is certainly something odd about these infinite sets, because they don’t behave as we might expect. Notice, for example, that the set S/2 is a subset of S, because S contains every entry in S/2, but it also contains ½. And yet S and S/2 are the same size! This counter-intuitive nature of infinite sets led to one of the great Infinite Monkey Cage arguments that took place between Brian and comedy producer John Lloyd in the form of Hilbert’s Grand Hotel paradox.
John Lloyd: Infinity plus one is just intellectual brain bending and I cannot see the use of it. Infinity is a word. That belongs to the wordy people like me and Robin. The point is you cannot place a numerical value to infinity and therefore you cannot add a plus one to it or a minus one.
Brian: There are either an infinite number of numbers or there aren’t. I don’t see what the problem is?
John: There aren’t an infinite number of numbers, because you can always have more than infinity and so infinity is a meaningless concept.
Series 9, Episode 4 (9 December 2013)
Hilbert’s Grand Hotel has infinitely many rooms, and they are all occupied. What happens if a further guest turns up unannounced? The guest in Room 1 can be moved into Room 2, the guest in Room 2 can be moved into Room 3, and so on, freeing up Room 1 for the new guest. There is always room in Hilbert’s Grand Hotel, even when it is full. We haven’t increased the size of the hotel, and yet we’ve accommodated another guest. Using the language above, we can say that the cardinality of the set of rooms in Hilbert’s Grand Hotel is the same as the cardinality of the set of guests. Notice that this implies that we can slot an infinite number of extra guests into Hilbert’s Grand Hotel, even when it is full. To see this, note that we could have moved the guest in Room 1 to Room 2, the guest in Room 2 to Room 4, the guest in Room 3 to Room 6, and so on. This frees up ALL the odd-numbered rooms, and since there are an infinite number of odd numbers, the Grand Hotel can now accommodate an infinite number of new guests. The reason for this strange state of affairs is that the number of odd rooms is not smaller than the number of even + odd rooms. Both sets have the same cardinality.
Figure 1
As an aside, there are sets with a different cardinality to the integers. Consider, for example, the set of infinite binary sequences; 0000000000…, 1111111111…, 0101010101… and so on. Cantor imagined laying these out in a vertical table, just as we did for our sets S and S/2, against the column of integers. Now imagine constructing a new binary sequence by changing all the digits in a diagonal line down the table; when there is a 1, swap it for a 0, and when there is a 0, swap it for a 1 (see Figure 1). The resulting sequence is not in the table; it can’t be the same as the sequence in the first row because the first digit has been flipped. It can’t be the same as the sequence in the second row because the second digit has been flipped, and so on. Every row will differ from the new sequence by at least one digit, claimed Cantor, because there are more infinite sequences of 1s and 0s than there are integers. A mathematician would say that the infinite set of binary sequences has a cardinality greater than the set of integers.
Infinity exists in mathematics, and as we have seen there are different sorts of infinity. We might ask whether there are infinities in the real world, and the answer is that we don’t know. The Universe may or may not be infinite in extent. The part of the Universe we can observe is certainly finite, and cosmologists refer to this as the Observable Universe. The most distant objects we can see are objects from which light has been able to travel during 13.8 billion years, which is the age of the Universe. You might be tempted to say that the Observable Universe is 27.6 billion light years across, but this is not correct because the Universe has been expanding throughout this time, and the objects have receded from us. Because we know how much the Universe has expanded, we can calculate the diameter of the Observable Universe today; it is just over 93 billion light years.
There are around 2 trillion galaxies in this observable sphere centred on the Earth, but this is very likely to be a small fraction of the entire Monkey Cage. Whether it is a finite subset of an infinite cage is an open question.
DEATH, WHERE IS THY PUDDING?
‘In many ways, Schrödinger’s Strawberry changed my life. Or death. Or whichever – it’s so hard to tell these days. All I know is the episode which featured that singular fruit gave me and my career a new lease of life. Or death. Oh god…’
Katy Brand, comedian