Marquette University

Department of Mathematics, Statistics and Computer Science

Wim Ruitenburg's Spring 2008 MATH025.1001

In this section we talk about mathematical infinity. The notes below are an extension on chapter 3 of the book.

Size matters

Informally, infinite sets look like very large finite sets. They are less alike. Below we look at finite sets and infinite sets, and their different natures.

Finity

We say that two finite collections are of equal size when we can pair their elements one by one. For example, a collection with elephants A, B, and C, written {A,B,C}, is considered of equal size as a collection of mice a, b, c, written {a,b,c}. They both have three elements. The number 3 symbolizes this size `property.' When we consider the total mass of the elephants versus the total mass of the mice, then the elephants give us more. However, when we talk about the size of collections, or sets, of objects, we just mean their number of objects without further interpretation. We can pair A with a, B with b, and C with c, and are done with it. The sets are of equal size. They both have 3 elements.
The state has about 5 million residents. For simplicity, let us assume that it is exactly 5 million. So the set of state residents has size 5 million, a very big number. Suppose a new person becomes resident in our state. The new set of residents now has size 5000001. In practice we would not notice the difference. Mathematically, however, the new set has a bigger size than the original one. No matter how we try to correspond elements of the original set with elements of the new set, the new set will always have a resident without correspondence to a resident of the original set.
The state has about 5 million residents. For simplicity, let us assume that it is exactly 5 million. So the set of state residents has size 5 million, a very big number. Suppose a person leaves the state. The new set of residents now has size 4999999. In practice we would not notice the difference. Mathematically, however, the new set has a smaller size than the original one. No matter how we try to correspond elements of the original set with elements of the new set, the original set will always have a resident without correspondence to a resident of the new set.

Countable infinity

In mathematics we also have infinite sets. The first main example is the collection of whole numbers, or natural numbers. We sometimes write N as symbol for this set. We can also describe N by listing the collection of its elements, like {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...}, where the three dots mean that we should continue without end. Just as for finite sets, we say that two collections are of equal size when we can pair their elements one by one.
Besides N, we can easily think of other infinite sets. For example, the collection of prime numbers. Let us write P for the set of primes. Then P can also be described by {2, 3, 5, 7, 11, 13, 17, ...}, where the reader is expected to understand how to continue this listing without end. Obviously, the set of primes has properly fewer elements than the set of natural numbers N. What is the size of P versus the size of N? Remarkably, we can pair the elements of N with the elements of its proper subcollection P:

0 1 2 3 4 5 6 7 8 ....

2 3 5 7 11 13 17 19 23 ....

So we can draw two interesting conclusions:
- There are as many whole numbers as there are primes.
- With infinite sets it is possible that a proper subcollection has the same size as the whole collection.

We now look at sets which seem bigger than N, but really are of the same size as N.

The set of pairs of natural numbers is written as N x N. Its elements look like (7,43) or (0,99) or (132,15). We can arrange the elements of this set as points of an infinite plane, like

....	....	....	....	....	....	....
(0,5)	(1,5)	(2,5)	(3,5)	(4,5)	(5,5)	....
(0,4)	(1,4)	(2,4)	(3,4)	(4,4)	(5,4)	....
(0,3)	(1,3)	(2,3)	(3,3)	(4,3)	(5,3)	....
(0,2)	(1,2)	(2,2)	(3,2)	(4,2)	(5,2)	....
(0,1)	(1,1)	(2,1)	(3,1)	(4,1)	(5,1)	....
(0,0)	(1,0)	(2,0)	(3,0)	(4,0)	(5,0)	....

We can one-one correspond these elements in the `plane' with the points of the `line' N as follows.

....	....	....	....	....	....	....
20	26	33	41	50	60	....
14	19	25	32	40	49	....
9	13	18	24	31	39	....
5	8	12	17	23	30	....
2	4	7	11	16	22	....
0	1	3	6	10	15	....

For example, 0 corresponds with (0,0); and 4 corresponds with (1,1); and 23 corresponds with (4,2); and so on. So the size of the `line' N of whole numbers is equal to the size of the `plane' N x N of pairs of whole numbers.

Given the result above, it is no longer a surprise that the `space' N x N x N of triples of whole numbers is also of the same size as the `line' of whole numbers N. A one-one correspondence between N x N x N and N can be constructed by putting together two one-one correspondences as follows:
- The first between N x N x N and N x N. Read triples like (2,0,5) as a combination of a point (2,0) in the `plane' N x N, plus a point 5 on the `line' N. Then use the one-one correspondence above to connect (2,0) with 3 while leaving 5 unchanged, giving the point (3,5) in the `plane' N x N.
- The second between N x N and N. Use the one-one correspondence above to connect (3,5) in the `plane' N x N with 41 on the `line' N.
Two correspondences can be put together into a single one-one correspondence between N x N x N and N. The composition will one-one correspond a triple like (2,0,5) with 41.
Let us write Q for the collection of rational fractions. That is, Q has elements like 2/3, -5/4, 71/40, -1/9, 5, 100/99, and so on. We know from the Pythagoreans that the fractions can be seen as a line of points. The line is dense in that it feels as if they fill the whole line with no room left for other elements. This is not true; the ancient Greeks already showed that some square roots are on the line, but are different from all rational fractions. Remarkably, the set of fractions Q has the same size as the set of natural numbers N. Here is an incomplete sketch of why this is true. Correspond all positive rational numbers with elements of the `plane' N x N as follows:
- Correspond 2/3 with (2,3)
- Correspond 71/40 with (71,40)
- Correspond 5 with (5,1)
- Correspond 100/99 with (100,99)
- And so on. So the positive fractions correspond with a subcollection of N x N. There is still plenty of room in N x N to also correspond 0 and the negative fractions.

Countable examples

Fill the barrel. This is a better and earlier version of the one on page 149 in the book. In a room there stands a barrel. Next to the barrel is a pile of marbles; an infinite collection of marbles, numbered 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and so on.
- At midnight, Ahab puts marbles numbered 0 and 1, in the barrel. At 1 in the morning, Jezebel takes the marble numbered 0, out of the barrel.
- At 6 , Ahab puts marbles numbered 2 and 3, in the barrel. At 6:30, Jezebel takes the marble numbered 1, out of the barrel.
- At 9 , Ahab puts marbles numbered 4 and 5, in the barrel. At 9:15, Jezebel takes the marble numbered 2, out of the barrel.
- At 10:30 , Ahab puts marbles numbered 6 and 7, in the barrel. At 10:37:30, Jezebel takes the marble numbered 3, out of the barrel.
- At 11:15 , Ahab puts marbles numbered 8 and 9, in the barrel. At 11:18:45, Jezebel takes the marble numbered 4, out of the barrel.
- And so on, getting closer and closer to noon.
At noon, the barrel is empty.
Hilbert Hotel. This is a better and earlier version of Hotel Cardinality on page 157 of the book. Suppose we are managers of a Hilbert Hotel, a hotel with infinitely many rooms, numbered 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and so on. The hotel is fully booked for the night. Then we get late new arrivals.
- A new guest shows up, begging for a room. Can we accommodate her?
  - Yes. We engage the intercom to all rooms, and ask all current guests to (simultaneously) move to the next room up. So the resident of room 0 moves to room 1. The resident of room 1 moves to room 2. The resident of room 2 moves to room 3. The resident of room 3 moves to room 4. The resident of room 4 moves to room 5. And so on. Now all current guests still have a room, while room 0 has become available for the latest arrival.
- Two more guests show up, begging for 2 rooms. Can we accommodate them?
  - Yes. You figure this one out yourself.
- Across the street is another Hilbert Hotel, also fully booked. Careless smoking leads to the other Hilbert Hotel to burn to the ground. Luckily all its former guests and their private belongings are saved. They come knocking on our door and beg for rooms. Can we accommodate them?
  - Yes. We engage the intercom to all rooms, and ask all current guests to move to the room with double their current room number. The resident of room 0 stays in room 0. The resident of room 1 moves to room 2. The resident of room 2 moves to room 4. The resident of room 3 moves to room 6. The resident of room 4 moves to room 8. And so on. Now all current guests still have a room, while the odd-numbered rooms 1, 3, 5, 7, 9, 11, 13, 15, 17, and so on, are available for the burned-out guests from across the street.
- Around the corner is Hilbert Street, an infinitely long street. Along Hilbert Street there are Hilbert Hotels numbered Hilbert Hotel 0, Hilbert Hotel 1, Hilbert Hotel 2, Hilbert Hotel 3, Hilbert Hotel 4, Hilbert Hotel 5, Hilbert Hotel 6, Hilbert Hotel 7, Hilbert Hotel 8, and so on. All are fully booked. An earthquake levels all hotels on Hilbert Street. What is the question? What is the answer?

Larger infinities

Let us write R for the set of all real numbers. So R contains all the points on the line. On the CD, and in the book on page 195, we see a one-one correspondence between the line R and the set of (all but one) points on a circle.
Are the set of natural numbers N and the set of real numbers R of equal size? During the 1870s, Georg Cantor showed that they are not. The proof uses the famous Cantor Diagonal Argument. The proof is by contradiction. Suppose there were a one-one correspondence between N and R. We can write it in a table like

N R

0 3.450581...

1 -473.109024...

2 89.648283...

3 0.592503...

4 -51.884601...

5 660.128462...

... ...

Now let us write the digits behind the decimal points as

450581...

109024...

648283...

592503...

884601...

128462...

...

where we boldface the digits `on the diagonal.' The boldface digits give a list of digits 408502... Now replace all occurrences of 0 in the list by 1, and replace all occurrences of 1, 2, 3, 4, 5, 6, 7, 8, or 9, by 0. So we get a new list of digits 010010... Now consider the special number 0.010010... It differs from the top number in the correspondence between N and R because they differ in the first digit behind the decimal point. The special number differs from the next number in the correspondence between N and R because they differ in the second digit behind the decimal point. The special number differs from the following number in the correspondence between N and R because they differ in the third digit behind the decimal point. And so on. So the special number does not occur on the right hand side in the correspondence between N and R above. So it is not really a one-one correspondence between N and R at all. Thus R has a truly bigger infinity than N.

All we need are sets?

Much like with Euclid and his Elements, some 2000 plus years ago, many mathematicians want a new foundations for mathematics which allow them to talk about their new theorems in less ambiguous ways or in more uniform ways. By the end of the 19th century, a new idea developed on how to do this: Set Theory. Just like with Elements, but at higher standards, set theory starts off with axioms, from which one rigorously derives all known theorems of our time.
It appeared that all that mathematicians really talk about were collections which we now call sets. From a mathematical perspective, only their elements matter. Mathematicians have ways to construct new sets, sometimes directly, sometimes from old sets. Here are some examples.
- The set N of natural numbers.
- Naturally, Q, and R are also sets.
- If A and B are sets, then we can also form the set A x B of all pairs (a,b) of elements. Recall how we introduced N x N and N x N x N earlier.
- The empty set.
- If A is a set, then so is its collection P(A) of all its subsets.
  - For example, let A be the set {0}. Then P(A) has elements {} (the empty subset of A) and {0} (the whole set A is a subset of A).
  - Let B be the set {0,1}. Then P(B) has elements {}, {0}, {1}, and {0,1}.
  - Let C be the set {0,1,2}. Then P(C) has elements {}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, and {0,1,2}.
  - Examples of elements of P(N) are the set P of all primes, the set of even numbers, the empty set, and {0,1,2,3}.
  - Remarkably, one can show that R and P(N) have the same size.
  - In the book, page 182, one shows that for all sets A, the size of P(A) is bigger than the size of A. We already knew this for finite sets. Now we also know it for infinite sets. So with the sequence N, P(N), P(P(N)), P(P(P(N))). P(P(P(P(N)))) and so on, we get a list of sets with larger and larger sizes of infinity.
- Sets themselves can be used as elements for making new sets. So {0,{0},N} is a set with three elements.
As far as mathematicians are concerned, each set is solely talking about what its elements are. So the set of even numbers is the same as the set of natural numbers divisible by 2 is the same as the set natural numbers that are multiples of 2.
In the early days of set theory development, some assumed that everything could be a set, as long as it could be named. For example,
- one would claim that if `x is a banana' were a well-defined statement, then we would have a set of all bananas, written {x : x is a banana}.
- The set of primes P can be defined as {x : x is a whole number such that x is prime}. The statement `3 is in {x : x is a whole number such that x is prime}' is true. The statement `4 is in {x : x is a whole number such that x is prime}' is false.
Shortly after 1900, Bertrand Russell considered the following set:
- {x : x is not in x}
Let us call this set R.Then it is either true that R is in R, or it is not true that R is in R. Assume that R is in R. Then, by the definition of R, we must have R is not in R. Contradiction. Our assumption is false. Assume that R is not in R. Then, by the definition of R, we must have R is in R. Contradiction. Our assumption is false. We have a paradox. Mathematicians, logicians, and philosophers, have worked for many years, and argued for many years, on the meaning of this paradox, and on how to avoid it.
Hilbert and others looked for ways to use mathematics itself to show that paradoxes and their likes can not occur in significant parts of mathematics. In 1931, Kurt Gödel proved that this was not really possible.

Last updated: April 2008
Comments & suggestions: wimr@mscs.mu.edu