Wim Ruitenburg's Spring 2017 MATH 1300-101

Marquette University

Department of Mathematics, Statistics and Computer Science

Wim Ruitenburg's Spring 2017 MATH 1300-101

Last updated: March 2017
Comments and suggestions: Email wim.ruitenburg@marquette.edu

Size matters

In this section we talk about mathematical infinity. 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 new set with elements of the original set, the new set always has 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.

Larger infinities

Let us write R for the set of all real numbers. So R contains all the points on the line. In class we saw 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? In 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?

You should have heard that school geometry is also known as Euclidean geometry. This refers to the Greek Euclid who around 300 BC wrote a famous book called Elements. For about 2000 years this book represents a foundations for mathematics. By the end of the 19th century 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. Their new ideas are now known as Set Theory. Georg Cantor is considered the founder of set theory, and derived some of its most unusual properties.
It appears that all that mathematicians really talk about are 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, the set Z of whole numbers, the set Q of fractions, and the set R of real numbers.
- 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 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} (1 element). Then P(A) has 2 elements, namely {} (the empty subset of A) and {0} (the whole set A is a subset of A).
  - Let B be the set {0,1} (2 elements). Then P(B) has 4 elements, namely {}, {0}, {1}, and {0,1}.
  - Let C be the set {0,1,2} (3 elements). Then P(C) has 8 elements, namely {}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, and {0,1,2}.
    Note that if X has n elements, then P(X) has 2^n elements.
  - The following even applies to infinite sets A: The size of P(A) is bigger than the size of A. 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.

The following are some extra details about sets.

Here is Cantor's proof that the size of P(B) is always bigger than the size of B. For assume that we have a function (like a correspondence) which assigns to each element b in B a set f(b) (that is what elements in P(B) are). Now consider the `diagonal' set D of exactly all x in B such that x is not in set f(x). So for all x we have that x is in D exactly when x in not in f(x). So if there is no b with D = f(b). Thus this set D in P(B) does not correspond with any f(b).
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.

Example Problems

We are given the set A = {a,b,c,d,e}, the set B = {1,2,3,4}, and the set C of all positive even numbers of value at most 10. Order these three sets by increasing size. Which ones are of equal size?
Show that the set Z = {..., -3, -2, -1, 0,1,2,3,...} of all whole numbers is of the same size as the set of even numbers E = {0,2,4,6,8,...}.
The beaches of Normandy have lots of grains of sand. Is the number of grains of sand on the beaches of Normandy less than the size of the set of natural numbers N, equal to the size of N, or larger than the size of N?
Suppose we have a pile with 1 peanut.
Next to it we have another pile, with 2 peanuts.
Next to that one, we have another pile, with 3 peanuts.
Next to that one, we have another pile, with 4 peanuts.
Next to that one, we have another pile, with 5 peanuts.
Next to that one, we have another pile, with 6 peanuts.
And so on. Do we have the same number of peanuts as there are natural numbers N = {0,1,2,3, 4,5,6,7,...}, or is the size of the peanut collection really bigger than the size of the collection of natural numbers?
What is Georg Cantor most famous for?
Show that the collection of real numbers with values between 0 and 1, is equal in size to the collection of real numbers with values between 0 and 5.
How many elements are in the set A = {0,{0},1,{{1},2}}?
Given the set B = {0,1,2,3,4}, how many elements are in its power set P(B)?