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Big, Bigger, Infinity, and a Bigger Infinity

Sybilla Beckmann

January 30, 2007

Are your students interested in big numbers and in the idea of finding ever bigger numbers?Children often like to challenge each other by finding greater and greater numbers: a million is big,2 million is bigger, a billion and a trillion are greater still. There could be millions of blades of grasson a lawn and there could be billions of grains of sand on a beach; there are billions of stars inthe universe. Then there is a mysterious number: a googol, which we can write as a 1 followed by100 zeros. This number is incomprehensibly large for us humans because it is even greater than thenumber of atoms in the universe. But still, there are greater numbers: a googol plus one is greater,

so is 2 googol, and then there is a googolplex, which would be written as a 1 followed by a googolzerosalthough nobody could ever write this number. Can you explain why not?Sometimes a contest of finding greater and greater numbers ends when somebody says infinity.

But is there really nothing bigger than infinity? What about infinity plus 1? Is that bigger than justinfinity, or is it really just the same infinity? What about infinity times infinity? Is it even bigger,or not? Are all infinities created equal, or are some more infinite than others? Surprisingly, themathematician Georg Cantor (1845 1918) discovered that there isnt just one infinity, but there aremany infinities, and some infinities are greater than others! But, the situation is more subtle thanyou might think, because infinity plus 1 and infinity times infinity are actually the same as justplain old infinity, as we will see. However, as we will also see, the infinity of the counting numbersis a smaller infinity than the infinity of all the decimal numbers, or even of all the decimal numbers

between 0 and 1. These were shocking discoveries for mathematicians, and this is deep and subtlemathematics.

Different infinities versus the same infinity

The key to understanding the idea of different infinities is first to say what it means for sets (orcollections of objects) to have the same number of members. If you have a collection of pennies andanother collection of nickels, how can you tell if you have the same number of pennies as nickels?You could count the pennies and count the nickels and see if the numbers are the same. But anotherway is to see if you can make a one-to-one correspondence between the pennies and the nickels. Ifevery penny can be paired with a nickel so that no nickels are left over, then the number of pennies

is the same as the number of nickels. In the same way, we will say that two infinite sets have thesame cardinality if there is a one-to-one correspondence between the two sets, in other words, if everymember of the first set can be paired with a member of the second set so that no member of thesecond set is left over.

Infinity plus 1, two times infinity, and infinity times infinity

So now lets think about infinity versus infinity plus 1. Are these the same or not? Lets be morespecific, lets consider the set of counting numbers, {1, 2, 3, 4, . . .} and take this set to represent

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(1,1) (2,1) (3,1) (4,1) (5,1) (6,1) . . .

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

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

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

(1,5) (2,5) . . .

(1,6) . . .

Figure 2: A Path Through the Ordered Pairs of Counting Numbers

are integers, and b is not zero. How many rational numbers are there? How many rational numbersare there between 0 and 1? Think about these questions before you read on.

There are infinitely many rational numbers and even infinitely many rational numbers between0 and 1. The list 1

2, 1

3, 1

4, . . . , for example, is an infinite list of rational numbers between 0 and 1.

Since every counting number is a rational number and since there are lots of rational numbers fillingin the number line between 0 and 1, 1 and 2, 2 and 3, and so on, it may seem that there are morerational numbers than counting numbers. But are the cardinalities of these sets different? Thinkabout whether this question may be related to any of the previous discussions before you read on.

The argument above, showing that the cardinality of the set of ordered pairs of counting numbersis the same as the cardinality of the counting numbers, can be used to show that the positive rationalnumbers have the same cardinality as the counting numbers. How? Roughly, we can view a pair(a, b) as representing the fraction a

b(so that (2, 3) represents 2

3, for example). So even though it may

seem like there are more fractions than counting numbers, there really arent. (Notice a technicality:(2, 3) and (4, 6) actually yield the same rational number since 2

3= 4

6, so the correspondence actually

isnt one-to-one, but even so, no positive rational number gets left out, and so there are at least asmany counting numbers as positive rational numbers.)

Lets consider another example. What about all the decimals between 0 and 1? Are thereinfinitely many such decimals? Think about this before you read on.

There are infinitely many decimals between 0 and 1, because, for example, the fractions 12

, 13

, 14

,. . . , can be represented as decimals. The list .1, .01, .001, .0001, .00001, . . . , is another infinite listof decimals, all of which lie between 0 and 1. Is the infinity of the decimals between 0 and 1 the sameinfinity as that of the counting numbers or not? In other words, can we put the set of all decimalsbetween 0 and 1 in one-to-one correspondence with the counting numbers or not?

Since the decimals between 0 and 1 lie within such a small range (none of them are even greaterthan 1!), it might seem that the infinity of them is smaller than the infinity of the counting numbers,but surprisingly, it is not. In fact, the infinity of the decimals between 0 and 1 is actually greaterthan the infinity of the counting numbers! To see why this is so, we will use a beautiful argument ofGeorg Cantors which shows that there cannot be a one-to-one correspondence between the countingnumbers and the set of decimals between 0 and 1.

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A game that explores Cantors idea (optional)

In order to explore Cantors idea, lets play a game. You will make a list of decimals (all between 0and 1) and I will show you more and more digits of a single decimal that I make. Your object is toshow that its possible to make a list that contains all the decimals between 0 and 1. My object isto make a decimal that isnt on your list. Ok, you go first. Write down a decimal you actuallyonly have to show me its first digit to the right of the decimal point. So you only have to show me

whats in the tenths place of your first decimal.Now its my turn. My decimal is .1 . . . if the digit in the tenths place of your decimal was not 1,

and its .2 . . . if your decimal has a 1 in the tenths place.Your turn. Write down a second decimal (between 0 and 1). This time you only have to show

me its first two digits to the right of the decimal point. Remember that you are trying to show thatyou can make a list with all the decimals between 0 and 1 in it, so you might want to try to get mydecimal on your list.

Now its my turn. Im going to put a 1 in the hundredths place if your second decimal doesnthave a 1 in the hundredths place and I put a 2 in the hundredths place if your second decimal doeshave a 1 in the hundredths place.

Your turn again. Write down a third decimal. This time you only have to show me its first threedigits to the right of the decimal point.

My turn again. Im going to put a 1 in the thousandths place if your third decimal doesnt havea 1 in the thousandths place and I put a 2 in the thousandths place if your third decimal does havea 1 in the thousandths place. Do you see the strategy I am using?

You might want to play this game with another person to get a feel for it.After playing this game a number of times, you should see that no matter how you pick your list

of decimals, I can arrange for my decimal not to be on your list. The strategy I used is essentiallyCantors brilliant diagonalization argument.

Cantors diagonalization argument

What is Cantors diagonalization argument and what does it have to do with different infinities?Cantors idea is to suppose at first that you could make a one-to-one correspondence between thecounting numbers and the decimals between 0 and 1. Then you would pair 1 with a decimal, pair 2with another decimal, pair 3 with yet another decimal, and so on. In other words, you would have alist of decimals, a first, a second, a third, and so on, such that every decimal between 0 and 1 is inthat list. Lets suppose for the moment that the list looks like this:

0.13476522 . . .0.93325192 . . .0.74285109 . . .

...

The heart of Cantors idea is to focus on the first digit of the first number, the second digit of thesecond number, the third digit of the third number, and so on:

0. 1 3476522 . . .

0.9 3 325192 . . .

0.74 2 85109 . . ....

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The point is that this list cant contain all the decimals between 0 and 1 because we can find adecimal that is guaranteed to be different from all of these numbers. Namely, make a decimal sothat the first digit to the right of the decimal point something different from 1, say 2, so that ourdecimal starts as

0.2 . . .

So no matter what comes next, our decimal is different from the first number in the list. Continuing

our decimal, we pick the digit in the hundredths place so that it is different from the digit in thehundredths place of the second decimal in the list, we pick the digit in the thousandths place so thatit is different from the digit in the thousandths place of the third decimal in the list, and so on. Forexample, our decimal could start out

0.211 . . .

The first digit, 2, is different from 1, the second digit, 1, is different from 3, the third digit, 1, isdifferent from from 2, so 0.211 . . . is definitely different from the first three decimals in the list above.Continuing in this way, we can make a decimal that is guaranteed to be different from every decimalin the list of decimals that supposedly contains all the decimals between 0 and 1. This is Cantorsdiagonalization argument, to produce a decimal that is different from all the decimals in a list by

considering the digits along the diagonal of the list.Now you might say: just add that different decimal you found to the list, then the list will becomplete. But the problem is that the list was supposed to have been complete from the start. Theargument above is a proof by contradiction. If you read Bill McCallums essay [title], then you sawanother proof by contradiction. In this case, we assumed that there was a one-to-one correspondencebetween the counting numbers and all the decimals between 0 and 1, and we showed that such a one-to-one correspondence would have to miss at least one decimal, which is a contradiction. Thereforethere cannot be a one-to-one correspondence between the counting numbers and the decimals between0 and 1.

So the cardinalities of the counting numbers and the decimals between 0 and 1 are not the same.In other words, the infinity of the decimals between 0 and 1 is a bigger infinity that the infinity of

the counting numbers!

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