2. Infinity and Induction

2.1. Countable Infinity

One of the more obvious features of the three number systems N, Z, and Q that were introduced in the previous chapter is that each contains infinitely many elements. Before defining our next (and last) number system, R, we want to take a closer look at how one can handle 'infinity' in a mathematically precise way. We would like to be able to answer questions like:

  1. Are there more even than odd numbers ?
  2. Are there more even numbers than integers ?
  3. Are there more rational numbers than negative integers ?
While most people would probably agree that there are just as many even than odd numbers, it might be surprising that the answer to the last two questions is no as well. All of the sets mentioned have the same number - albeit infinite - of elements. The person who first established a rigorous 'theory of the infinite' was G. Cantor.

The basic idea when trying to count infinitely large (or otherwise difficult to count) sets can roughly be described as follows:

This simple idea of matching two sets element by element is the basis for comparing two sets of any size, finite or infinite. Since 'matching elements from one set with those in another set ' seems related to the concept of a function, we have arrived at the following definition:
Definition 2.1.1: Cardinality
  • Let A and B be two sets. We say that A and B have the same cardinality if there is a bijection f from A to B. We write card(A) = card(B).
  • If there exists a function f from A to B that is injective (i.e. one-to-one) we say that card(A) card(B).
  • If there exists a function f from A to B that is surjective (i.e. onto) we say that card(A) card(B).
Please explain carefully what this definition has to do with the above idea of counting students and chairs?
Examples 2.1.2:
  • We can now answer questions similar to the ones posed at the beginning:
    1. Let E be the set of all even integers, O be the set of odd integers. Then card(E) = card(O). What is the bijection ?
    2. Let E be the set of even integers, Z be the set of all integers. Again, card(E) = card(Z). Can you find the bijection ?
    3. Let N be the set of natural numbers, Z be the set of all integers. Which set, if any, has the bigger cardinality ?
Definition 2.1.3: Countable and Uncountable
  If a set A has the same cardinality as N (the natural numbers), then we say that A is countable. In other words, a set is countable if there is a bijection from that set to N.

An alternate way to define countable is: if there is a way to enumerate the elements of a set, then the set has the same cardinality as N and is called countable.

A set that is infinite and not countable is called uncountable.

The second part of this definition is actually just rephrasing of what it means to have a bijection from N to a set A: By the above examples, the set of even integers, odd integers, all positive and negative integers are all countable.

Note that there is a difference between finite and countable, but we will often use the word countable to actually mean countable or finite (even though it is not proper). However, here is a nice result that distinguishes the finite from the infinite sets:

Theorem 2.1.4: Dedekind Theorem
  A set S is infinite if and only if there exists a proper subset A of S which has the same cardinality as S.

Examples 2.1.5:
  • Use Dedekind's Theorem to show that the set of integers Z and the interval of real numbers between 0 and 2, [0, 2], are both infinite (which is of course not surprising).
The surprising fact when dealing with countably infinite sets is that when combining two countable sets one gets a new set that contains no more elements than each of the previous sets. The next result will illustrate that.
Proposition 2.1.6: Combining Countable Sets
  • Every subset of a countable set is again countable (or finite).
  • The set of all ordered pairs of positive integers is countable.
  • The countable union of countable sets is countable
  • The finite cross product of countable sets is countable.

Think about these propositions carefully. It seems to be contrary to ones beliefs. To see some rather striking examples for the above propositions, consider the following:

Examples 2.1.7:
  • The set of all rational numbers is countable.
  • The collection of all polynomials with integer coefficients is countable. To prove this, follow these steps:
    1. Show that all polynomials of a fixed degree n (with integer coefficients) are countable by using the above result on finite cross products.
    2. Show that all polynomials (with integer coefficients) are countable by writing that set as a countable union of countable sets.

A nice example that illustrates how difficult and counter-intuitive the concept of infinity can be is provided by the story of Hilbert's Hotel. This particular version of Hilbert's Hotel has been created by Jillian Gaglione, a former Seton Hall student and avid Yankee (a New York baseball team) fan - with apologies to any Red Sox (a Boston baseball team) fan.

Example 2.1.8: Hilbert's Hotel
  Imagine two hotels in Boston, Massachusetts; the Holiday Inn, which has three hundred rooms and Hilbert's Hotel, which contains a countable infinite number of rooms. The New York Yankees are leaving town after defeating the Red Sox, again! However, their tour bus breaks down and they are forced to get off the bus and stay overnight in Boston. The Yankees, excluding their pitching coach, decide to walk to the Holiday Inn only to find that all three hundred rooms are already occupied. At the same time the pitching coach decides to walk over to Hilbert's Hotel because he heard it was a nicer hotel. When the pitching coach arrives at Hilbert's Hotel, he discovers that all of the rooms are taken as well. However, the manager of the hotel, a Mr. Hilbert, is an extremely big Yankee fan and says, after some thought, that he could accommodate one more person.
If all of the rooms are occupied, how is the manager of Hilbert's Hotel going to find a room for the pitching coach?
After the pitching coach settles into his room, he calls his two best friends, the right fielder and the shortstop, to tell them to come over to Hilbert's Hotel for a room. When they arrive at Hilbert's Hotel the manager informs them that all the rooms are still occupied, but, once again, he can find accommodations for two more people.
If all of the rooms are still occupied, how can the manager of Hilbert's Hotel find room for two more players?
The Yankees soon find out that Hilbert's Hotel was accommodating some members of their team and the manager of the hotel finds himself trying to find rooms for the rest of the members. There were fifty-eight people that need to find a room and still none of the guests have checked out of their rooms. The manager of the hotel says not to worry because he could find rooms for the remaining members.
If every room is occupied, how can the manager find enough rooms for the entire team?
Throughout the night, rumors spread that the Yankees are staying in Hilbert's Hotel. Many fans from the surrounding area travel to Hilbert's Hotel to book a room. When Hilbert looks outside his hotel, he sees a never-ending line of people that wish to stay for the night. But Hilbert wonders how he could find rooms for a countable infinite number of guests ... then he looks outside the hotel again and reassurs the people waiting in line that everyone will soon have a room for the night.
How could the manager of the hotel accommodate a countable infinite number of guests?
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