The Beginning of Infinity: Explanations That Transform the World (29 page)

The beginning of infinity – the rooms in Infinity Hotel

Now imagine that Infinity Hotel is fully occupied. Each room contains one guest and cannot contain more. With finite hotels, ‘fully occupied’ is the same thing as ‘no room for more guests’. But Infinity Hotel always has room for more. One of the conditions of staying there is that guests have to change rooms if asked to by the management. So, if a new guest arrives, the management just announce over the public-address system, ‘Will all guests please move immediately to the room numbered one more than their current room.’ Thus, in the manner of the first illustration in this chapter, the existing occupant of room 1 moves to room 2, whose occupant moves to room 3, and so on. What happens at the last room? There is no last room, and hence no problem about what happens there. The new arrival can now move into room 1. At Infinity Hotel, it is never necessary to make a reservation.

Evidently no such place as Infinity Hotel could exist in our universe,
because it violates several laws of physics. However, this is a
mathematical
thought experiment, so the only constraint on the imaginary laws of physics is that they be consistent. It is
because
of the requirement that they be consistent that they are counter-intuitive: intuitions about infinity are often illogical.

It is a bit awkward to have to keep changing rooms – though they are all identical and are freshly made up every time a guest moves in. But guests love staying at Infinity Hotel. That is because it is cheap – only a dollar a night – yet extraordinarily luxurious. How is that possible? Every day, when the management receive all the room rents of one dollar per room, they spend the income as follows. With the dollars they received from the rooms numbered 1 to 1000, they buy complimentary champagne, strawberries, housekeeping services and all the other overheads,
just for room 1
. With the dollars they received from the rooms numbered 1001 to 2000, they do the same for room 2, and so on. In this way, each room receives several hundred dollars’ worth of goods and services every day, and the management make a profit as well, all from their income of one dollar per room.

Word gets around, and one day an infinitely long train pulls up at the local station, containing infinitely many people wanting to stay at the hotel. Making infinitely many public-address announcements would take too long (and, anyway, the hotel rules say that each guest can be asked to perform only a finite number of actions per day), but no matter. The management merely announce, ‘Will all guests please move immediately to the room whose number is double that of their current room.’ Obviously they can all do that, and afterwards the only occupied rooms are the even numbered ones, leaving the odd-numbered ones free for the new arrivals. That is exactly enough to receive the infinitely many new guests, because there are exactly as many odd numbers as there are natural numbers, as illustrated overleaf:

There are exactly as many odd numbers as there are natural numbers.

So the first new arrival goes to room 1, the second to room 3, and so on.

Then, one day, an
infinite number
of infinitely long trains arrive at the station, all full of guests for the hotel. But the managers are still unperturbed. They just make a slightly more complicated announcement, which readers who are familiar with mathematical terminology can see in this footnote.
*
The upshot is: everyone is accommodated.

However, it
is
mathematically possible to overwhelm the capacity of Infinity Hotel. In a remarkable series of discoveries in the 1870s, Cantor proved, among other things, that not all infinities are equal. In particular, the infinity of the continuum – the number of points in a finite line (which is the same as the number of points in the whole of space or spacetime) – is much larger than the infinity of the natural numbers. Cantor proved this by proving that there can be no one-to-one correspondence between the natural numbers and the points in a line: that set of points has a higher order of infinity than the set of natural numbers.

Here is a version of his proof – known as the
diagonal argument.
Imagine a one-centimetre-thick pack of cards, each one so thin that there is one of them for every ‘real number’ of centimetres between 0 and 1. Real numbers can be defined as the decimal numbers between those limits, such as 0.7071. . ., where the ellipsis again denotes a continuation that may be infinitely long. It is impossible to deal out
one of these cards to each room of Infinity Hotel. For suppose that the cards
were
so distributed. We can prove that this entails a contradiction. It would mean that cards had been assigned to rooms in something like the manner of the table below. (The particular numbers illustrated are not significant: we are going to prove that real numbers cannot be assigned in
any
order.)

Cantor’s diagonal argument

Look at the infinite sequence of digits highlighted in bold – namely ‘
6996
. . .’. Then consider a decimal number constructed as follows: it starts with zero followed by a decimal point, and continues arbitrarily, except that each of its digits must differ from the corresponding digit in the infinite sequence ‘
6996
. . .’. For instance, we could choose a number such as ‘0.5885. . .’. The card with the number thus constructed cannot have been assigned to any room. For it differs in its first digit from that of the card assigned to room 1, and in its second digit from that of the card assigned to room 2, and so on. Thus it differs from all the cards that have been assigned to rooms, and so the original assumption that all the cards had been so assigned has led to a contradiction.

An infinity that
is
small enough to be placed in one-to-one correspondence with the natural numbers is called a ‘
countable
infinity’ – rather an unfortunate term, because no one can count up to infinity. But it has the connotation that every
element
of a countably infinite set could in principle be reached by counting those elements in some suitable order. Larger infinities are called
uncountable
. So, there is an uncountable infinity of real numbers between any two distinct limits.
Furthermore, there are uncountably many
orders
of infinity, each too large to be put into one-to-one correspondence with the lower ones.

Another important uncountable set is the set of
all logically possible reassignments
of guests to rooms in Infinity Hotel (or, as the mathematicians put it, all possible
permutations
of the natural numbers). You can easily prove that if you imagine any one reassignment specified in an infinitely long table, like this:

Specifying one reassignment of guests

Then imagine all possible reassignments listed one below the other, thus ‘counting’ them. The diagonal argument applied to this list will prove that the list is impossible, and hence that the set of all possible reassignments is uncountable.

Since the management of Infinity Hotel have to specify a reassignment in the form of a public-address announcement, the specification must consist of a finite sequence of words – and hence a finite sequence of characters from some alphabet. The set of such sequences is countable and therefore infinitely smaller than the set of possible reassignment. That means that only an infinitesimal proportion of all logically possible reassignments can be specified. This is a remarkable limitation on the apparently limitless power of Infinity Hotel’s management to shuffle the guests around.
Almost all
ways in which the guests could, as a matter of logic, be distributed among the rooms are unattainable.

Infinity Hotel has a unique, self-sufficient waste-disposal system. Every day, the management first rearrange the guests in a way that ensures that all rooms are occupied. Then they make the following announcement. ‘Within the next minute, will all guests please bag their trash and give it to the guest in the next higher-numbered room. Should you
receive
a bag during that minute, then pass it on within the
following half minute. Should you receive a bag during that half minute, pass it on within the following quarter minute, and so on.’ To comply, the guests have to work fast – but none of them has to work
infinitely
fast, or handle infinitely many bags. Each of them performs a finite number of actions, as per the hotel rules. After two minutes, all these trash-moving actions have ceased. So, two minutes after they begin, none of the guests has any trash left.

Infinity Hotel’s waste-disposal system

All the trash in the hotel has disappeared from the universe. It is
nowhere
. No one has
put
it ‘nowhere’: every guest has merely moved some of it into another room. The ‘nowhere’ where all that trash has gone is called, in physics, a
singularity
. Singularities may well happen in reality, inside black holes and elsewhere. But I digress: at the moment, we are still discussing mathematics, not physics.

Of course, Infinity Hotel has infinitely many staff. Several of them are assigned to look after each guest. But the staff themselves are treated as guests in the hotel, staying in numbered rooms and receiving exactly the same benefits as every other guest: each of them has several other staff assigned to their welfare. However, they are not allowed to ask those staff to do their work for them. That is because, if they all did this, the hotel would grind to a halt. Infinity is not magic. It has logical rules: that is the whole point of the Infinity Hotel thought experiment.