12. The Infinity Paradox:
Hilbert’s Hotel
The question:
Think of a fully booked hotel. No new guests can be accommodated. Now think of a huge hotel. It too is fully booked. Same problem. Now think of a hotel with an infinite number of rooms; let’s call it Hilbert’s Hotel. It is fully booked with an infinite number of guests. Can another guest be accommodated? Can many guests be accommodated? No and No. After all, the definition of ‘fully booked’ is that there are no more rooms available.
Correct?
The paradox:
Wrong! Let’s say all guests, an infinite number of them, have been assigned their rooms: guest number 1 to room 1, guest number 2 to room 2, guest number 3 to room 3, and so on and on. Now a new guest arrives, is there room for her? Surprise, surprise: there is.
All is settled but then even more unexpected guests arrive. In fact, an infinite number of new guests arrive. Is there room for all of them? In addition to the infinite number already there? Surprise, surprise: there is.
Background:
The examples stem from lectures that the mathematician David Hilbert (1862 – 1943) delivered at the University of Göttingen in Germany in the 1920s.
Göttingen was the nerve center of mathematics during the first decades of the twentieth century until Jewish scientists were chased away, deported or killed by the Nazis. Before its demise, budding mathematicians from all over the world flocked to this temple whose undisputed high priest was David Hilbert. He was one of the two last persons who knew everything there was to know about mathematics at their time. (The second was Henri Poincaré of France.) By now, the discipline has become so vast that nobody knows anything except his or her tiny specialty.
Though the concept of infinity had already been discussed by the ancient Greeks and Indians, it was usually in the realm of philosophy. The polymath Gottfried Wilhelm Leibniz speculated about it in the seventeenth century, but in proper mathematics, the notion was still awaiting a rigorous definition at the turn of the twentieth century. Hilbert based his allegory on the ideas of Georg Cantor (1845 – 1918), the inventor of set theory.
Finite sets are collections of objects, like the books in your library or the sweaters in your cupboard. They are quite simple and kids studying ‘modern mathematics’ may encounter the theory of finite sets in high school. More interesting sets contain an infinite number of objects, or elements, like the set of natural numbers (1,2,…, 17,…), the set of rational numbers (i.e., fractions like ½, 2/3 or 21/57), the set of real numbers (3.1415 or 2.71 etc.). The question posed at the outset of this chapter asks what one gets when one or more elements are added to such sets, or when several infinite sets are combined.
Dénouement:
How can the additional guests be accommodated at Hilbert’s Hotel and what does that mean for the concept of infinity?
The manager of the Hilbert’s Hotel is a very resourceful person. When the new guest arrives at the fully booked Hilbert’s Hotel, the hotel manager politely asks guest number 1 to move to room 2, guest number 2 to move to room 3, guest number 3 to room 4, and so on and on. Quite generally, guest number N is asked to move to room N+1. Now, lo and behold, room 1 is free and the new guest can be accommodated in it. Problem solved.
The guests are about to go to sleep in their newly assigned rooms, when, all of a sudden, new guests arrive…infinitely many! The quick-witted manager has no problem. “No problem,” he proclaims and asks guest number 1 to move to room 2, guest number 2 to move to room 4, guest number 3 to move to room 6, and so on, and so on. Put concisely, guest number N is asked to move to room 2N. Then, while the old guests, infinitely many, settle down in their even numbered rooms, the newly arrived guests, also infinitely many, are invited to move in to the odd numbered rooms.
Of course, the hotel manager can repeat the process with one more, two more, many more, infinitely more new guests. The story illustrates the elusive concept of infinity. Add 1 to infinity and the result is still infinity. And two or three to infinity…same thing. Two times infinity is also infinity. And infinitely many times infinity is….also infinity.
Technical Supplement:
The key mathematical concept when speaking of infinity in set theory is ‘cardinality’. The word describes a set’s mightiness. The infinitely large set of odd numbers, for example, has the same cardinality as the set of natural numbers. The number of rational numbers also has the same cardinality as the natural numbers. This may surprise for the following reason:
While the set of natural numbers consists of the elements
1, 2, 3, 4, ……all the way to infinity,
the set of rational numers contains the elements
1/1, 1/2, 1/3, 1/4, …. all the way to infinity
and
2/1, 2/2, 2/3, 2/4, …. all the way to infinity
and
3/1, 3/2, 3/3, 3/4, …. all the way to infinity
and
…
9/1, 9/2, 9/3, 9/4, …. all the way to infinity
…
…
all the way to infinity.
Naively, one may think that there are infinitely times as many rational numbers as there are natural numbers. Well that’s wrong. The rational numbers, like the natural numbers, are countable, from one to … infinity.
But how can the rational numbers be counted? After all, in the above table we see that the set of natural numbers has only one row with infinitely many numbers, while the set of rational numbers contains infinitely many rows, each with infinitely many numbers. Well, Georg Cantor devised an ingenious trick to count the rational numbers. He enumerated them diagonally as follows:
1/1, 1/2, 1/3, 1/4, …..
2/1, 2/2, 2/3, 2/4, ….
3/1, 3/2, 3/3, 3/4, …
Hence, rational numbers can be counted and therefore form what is called a ‘countably infinite’ set. Thus, Cantor proved that the set of natural numbers and the set of rational numbers have the same cardinality. Both contain a ‘countably infinite’ number of elements.
Are there sets that contain ‘uncountably infinite’ numbers of objects? Yes there are because not all infinities are equal.
Take, for example, the set of rational numbers. These numbers are of the form 5.31641…. usually with never-ending tails of decimals. Even Georg Cantor could not devise a method to count them all. We, therefore, say that the set of real numbers contains an ‘uncountably infinite’ number of elements. Hence it has a higher cardinality than the set of natural numbers or the set of rational numbers.
* * *
Georg Cantor’s life was tragic. Many of his contemporaries did not take him seriously. In fact, he was ridiculed by several colleagues, which is most probably what drove him to depair and led to numerous bouts of depression. After the sudden death of a son, he had to be hospitalized once again. He continued to suffer from chronic depression throughout his adult life and died from heart failure in an insane asylum.
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