Alex's Adventures in Numberland - Alex Bellos [172]
Now let’s translate what we have learned with the Hilbert Hotel into some symbolic mathematics:
When one person was found a room, this was the equivalent of showing that 1 + =
When a countably infinite number of people could be found a room, we saw that × =
When a countably infinite number of buses, each containing a countably infinite number of passengers could be found a room, this revealed that × =
These rules are what we expect from infinity: add infinity to infinity and we get infinity, multiply infinity by infinity and we get infinity.
Let’s stop here a second. We have already reached an amazing result. Look again at the table of seats and buses. Consider each thenn denoted m/n as the fraction . The table, when extended infinitely, will cover every single positive fraction, since the positive fractions can also be defined as for all natural numbers m and n. For example, the fraction will be covered on the 5628th row and 785th column. The zigzag counting method that counted every passenger in every bus can, therefore, also be used to count every positive fraction. In other words, the set of all positive fractions and the set of natural numbers have the same cardinality, which is . It seems intuitive that there should be more fractions than there are natural numbers, since between any two natural numbers there is an infinite number of fractions, yet Cantor showed that our intuition is wrong. There are as many positive fractions as there are natural numbers. (In fact, there are as many positive and negative fractions as there are natural numbers, since there are positive fractions and negative fractions and, as seen above, × = .)
We can appreciate how strange this result is by considering the number line, which is a way of understanding numbers by considering them as points on a line. Below is a number line starting at 0 and heading off towards infinity.
Every positive fraction can be considered as a point on this number line. From a previous chapter, we know that there is an infinite number of fractions between 0 and 1, as there is between 1 and 2, or between any two other numbers. Now imagine holding a microscope up to the line so that you can see between the points representing the fractions and . As we also showed earlier, there is an infinite number of points representing fractions between these two points. In fact, wherever you place your microscope on the line, and however tiny the interval between two points that your microscope can see, there will always be infinitely many points representing fractions in this interval. Since there are infinite numbers of points representing fractions wherever you look, it comes as a bewildering surprise to realize that it is, in fact, possible to count them in an ordered list that will cover every single one without exception.
Now for the big event: proof that there is a cardinality larger than . Back we go to the Hilbert Hotel. On this occasion, the hotel is empty when an infinite number of people show up wanting rooms. But this time the travellers have not come in buses; they are in fact a rabble, with each wearing a T-shirt displaying a decimal expansion of a number between 0 and 1. No two people have the same decimal expansion on their chest and every single decimal expansion between 0 and 1 is covered. (Of course, the decimal expansions are infinitely long and so the T-shirts would need to be infinitely wide to display them, yet since we have suspended our disbelief in order to imagine a hotel with an infinite number of rooms, I figure it is not asking too much to envisage these T-shirts.)
A few of the arrivals charge into reception and ask if there is a way the hotel can accommodate them. For the receptionist to achieve this all he needs to do is find a way of listing every single decimal between 0 and 1, since once he has listed them, he can assign them rooms. This seems like a fair