The notion of infinity is one of those concepts that seems beyond the realm of our understanding. Endless, forever; ideas that at first seem easy to understand, disintegrate upon closer examination. As a quick glimpse into the strange world of infinity, let’s take a look at that old mathematical standby, the number line.
The number line is a visual representation of “all the numbers” arranged in order on a line. Zero is in the middle, negative numbers go down the left side and positive numbers go down the right side. ½ is represented by a point between 0 and 1, “-pi” is represented by a point between -3.13 and -3.15 and so on. Before we go on, there are a few things to keep in mind about the number line.
The points on the number line have no real “size”. This is because the number line is infinitely divisible. Two points will never “touch” because there will always be another point in between them. It is helpful to think of the points on the number line not as dots but rather as intersections of other lines. Draw a line through the number line and it will cross at one point only. Since both lines are one dimensional constructs with no width, the zero dimensional point at which they meet will take up no space.
This leads to the question: If the number line is made up of points that take up no space, how many points will it take to “completely fill up” the number line? The answer as it turns out is not just “infinity” but rather “more than infinity”. Or more technically, the answer lies in there being more than one kind of infinity. To see this, let’s keep looking at the number line, but zoom out a bit and look at just the natural numbers (1, 2, 3, 4… but not 0).
It seems obvious that if you start at 1 and start counting, you will never reach a highest number. This is infinity. So if the natural numbers are infinite, we will use this fact and the natural numbers themselves to see what we can learn about “larger” groups of numbers.
The first “larger group” of numbers would be the whole numbers (all the natural numbers with 0 added). In this case the group is “one” larger, it has one more number than the natural numbers. Is it really larger? The key to answering the question of “what is infinity plus 1?” (the age old trump card of children the world over) lies in the concept of sets.
The set of all natural numbers can be thought of as the collection of each individual number from one to infinity. It is expressed like this: {1, 2, 3, 4,…}. So the set of all whole numbers would be {0, 1, 2, 3, 4,…}. Now, starting with the first term of our new set of whole numbers, let’s assign it one term from the set of natural numbers. 0 gets paired with 1, 1 with 2, 2 with 3, 3 with 4 and so on. By continuing this pairing all the way to infinity we can see that there exists a “one to one correspondence” between the two sets and therefore they contain the same amount of numbers! Remember, it’s no fair yelling “but the set of whole numbers will have one more term!” There is no last number in infinite sets, so there is no such thing as one extra term.
Let’s take that idea of sets and one to one correspondence and apply it to our next step on the number line, the integers (all the whole numbers (0, 1, 2, 3,…) and all the negative numbers out to negative infinity).
When we added the whole numbers it was easy enough to add only one item to our set and see that it was still in one to one correspondence with the set of natural numbers. But can we do the same thing with the negative numbers? After all {-1, -2, -3, -4…} seems like it would be exactly the same size as {1, 2, 3, 4,…} so when we add them together we get infinity + infinity. Surely this will result in a number greater than the set of natural numbers that we started with right?
As it turns out it does not. We can use roughly the same method that we used to show a one to one correspondence between the natural numbers and the whole numbers. All that must be done is to find a way to order the integers so that they can be paired up with the natural numbers. This time there is no beginning to our set (since it goes from negative infinity to positive infinity) so it is not as simple to line up each item in the set of integers with the set of natural numbers.
Still the way to order the terms becomes easy enough if we “rearrange” the integers into the following set: {0, 1, -1, 2, -2, 3, -3,…}. Now, we can see how these can easily be put into a one to one correspondence with the set of natural numbers and thus infinity + infinity = infinity!
Now it is time to include the fractions, thus creating the set of all rational numbers (all integers and all the fractions in between them–anything from ½ to 17/10375). So how many fractions are between 0 and 1? Infinity (consider the famous infinite sequence: 0, ½, ¾, 7/8, 15/16, 31/32… as one example of this). And how many spaces equal to the interval between 0 and 1 are there on the number line? Again, Infinity. So to get the amount of fractions on the entire number line we are dealing with the question of infinity times infinity, the answer to which must be greater than the set of all natural numbers right?
Again, the answer is no. Using the above techniques we can again show a one to one correspondence between the set of all rational numbers and the set of all natural numbers. This time it takes a bit more work than it did with the integers to show this correspondence but it can still be done. Create the following table with columns and rows that extend into infinity according to the pattern they are constructed in:
1/1 1/2 1/3 1/4 1/5 1/6…
2/1 2/2 2/3 2/4 2/5 2/6…
3/1 3/2 3/3 3/4 3/5 3/6…
4/1 4/2 4/3 4/4 4/5 4/6…
5/1 5/2 5/3 5/4 5/5 5/6…
etc…
It should be clear that this table will contain every possible fraction. 23/187,053,2343557? Just go down to the 23rd row and over to the 187,053,2343557 th column and there it is. Now, starting at the top left, order the terms in a winding orderly fashion so that you get the set:{1/1, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, 2/3, 3/2, 4/1, 5/1, 4/2, 3/3,…}. Take out the excess repeat numbers (2/2, 3/3, 4/2, etc) and you once again have a set of numbers that can be shown to have a one to one correspondence to the set of all natural numbers, and thus, infinity times infinity = infinity!
Before we move on to our next category of numbers let’s look at what we know so far. First, if you can establish a one to one correspondence between a set and the set of natural numbers then you can show that your set is equivalent to infinity. An interesting consequence of this is the fact that the set of all even numbers (or primes, or numbers ending in 1337) is equivalent to the set of all natural numbers even though those sets are contained within the set of all natural numbers!
I should also point out that I keep just using the general term “infinity” for all these sets, but technically they are called “countable” infinities (which of course does not mean that you can really count them, only that they can be put neatly in to a one to one correspondence with the set of all natural numbers). So are there then also “uncountable” infinities? Will we ever get an infinity “bigger” than the set of natural numbers?
To answer these questions let’s see what happens when we add the irrational numbers to our number line. In order to understand what irrational numbers are, we should take another look at the fractions. Every fraction will always either terminate or end in a repeating decimal. 1/2 = 0.5 1/3 = 0.33333… Again, these fractions (and their decimal expansions) are rational numbers. Any decimal that never terminates or repeats is an irrational number. These irrationals can be of two types, the algebraic irrational numbers and the transcendental irrational numbers.
The algebraic irrationals are basically irrational numbers that are the solutions to algebraic equations (I should add that all rational numbers are also algebraic). The square root of 2 is an algebraic irrational since it is the solution to the equation: x2 – 2 = 0 and its decimal expansion neither repeats nor terminates. Since the set of all polynomials of degree n with integer coefficients (basically equations like X+1=0 and 3×11 + 12×4 + x2 = 0) is countable then the rational numbers plus the irrational algebraic numbers is also countable.
To prove the countability of all polynomials (again, x2 – 3x + 2 is an example of a polynomial) we will start by looking at the first degree polynomials ( x-2, 6x+3, etc)(x2 – 3x + 2 is a degree “2” polynomial because of the x2, and so on)(0 degree polynomials are just the integers). Each first degree polynomial can be expressed as an ordered pair of integers, (1, -2) for x-2 and (6, 3) for 6x+3. Using the same technique we just used to prove the countability of the rationals, we can prove that these sets of pairs are also countable (just construct a table with the ordered pairs and then put them in order using diagonal swipes). From there it is a simple matter to add the second degree polynomials and show the countability of their coefficients (n1, n2, n3) and so on out to degree n. We will now turn to the fundamental theorem of algebra (a truncated version anyway) which for the purposes of this proof basically says for every polynomial of degree n there exist n unique real solutions (this is leaving a bit out, but should suffice for these purposes…we’ll just pretend everything that gets left out is imaginary). Since there are a countable number of polynomials and they have soloutions equal to their degree, there are a countable number of solutions! So as strange as it may seem, even the set of all polynomial equations is countable (and by extension, the irrational algebraic numbers as well!)
This leaves us with the transcendental numbers. These are numbers with non-terminating and non-repeating decimal expansions that are not a solution to a polynomial equation. This seems like a pretty narrow definition, and indeed, there are few well known transcendental numbers. The two most common are “pi” and “e”, but while there are many ways of generating transcendental numbers, there are few that are really in the spotlight like such number stalwarts as “1/2”, “0”, “-666”, “10” and “the square root of two”.
Before we look at the countability of the transcendental numbers, let’s check back in on the current state of our number line. By now we have all the rational numbers and even all the algebraic irrationals added to the number line. It’s getting pretty dense. Between every two numbers there exists another number and so on, down to (a countable) infinity. There doesn’t seem to be room for much more on our number line, so let’s go back and see where the seemingly uncommon transcendental numbers fit.
This brings us to the final (and definitely coolest) proof of this post, Cantor’s (mathematician who did more for the concept of infinity than all the others combined) diagonal proof. We are now working with the real numbers (the rationals and transcendental and algebraic irrationals together) and need a way to show that they are countable. To start, let’s make a list of all the real numbers like so:
1st real number – X1.a1a2a3a4….
2nd real number – X2.b1b2b3b4…
3rd real number – X3.c1c2c3c4…
4th real number – X4.d1d2d3d4…
5th real number – X5.e1e2e3e4…
.
.
.
nth real number
The numbers to the right of the decimal (the lower case letters) go on forever (even a number like “2” is written 1.99999… for these purposes), and the X’s also go on forever. Now, all we have to do is create a number that is not on this list and we can prove that the real numbers are uncountable. This can be done by looking at the first real number and taking its first digit X1 and subtracting one (or making it 9 if it is a 0). Then we take the second digit of the second real number and do the same thing, continuing with the third digit of the third real and fourth digit of the fourth real etc. Thus we turn the number X1.b1c2d3e4…into a new number (lets just call it X.abcd…) that is not on the list! If you are not sure, take a look at it again. For every real number there is one digit that is guaranteed to be different, for our new number was constructed by taking one digit from each real number and subtracting one (or changing a zero to nine). Also, you can not just add this new number to your list, because it is a simple matter to just apply Cantor’s diagonal proof again to generate a new number that is not on the list. Thus, we have finally found our set of numbers that is bigger (though such a notion can’t technically apply in infinite cases) than infinity, and in fact, we can say that the real numbers are uncountable!
So what are the implications of this? First, since the natural, whole, integer, rational and algebraic irrationals are countable, it is the transcendentals that are uncountable. This now leads to the frightening conclusion that our supposedly dense number line is actually made up all but entirely of transcendentals since there are an uncountably infinite amount of transcendentals between any two rational or algebraic irrational numbers no matter how close together they are! Quite a showing from a type of number that sees such infrequent use (barring e and pi of course)!
Perhaps the strangest thing we can take from all this is that something as simple as the smallest part of a straight line can contain within it something greater than infinity. Just think what the surface of an infinite plane, or even the area of everything contains! (Actually…they contain exactly the same amount of points as can be found between any two points on the number line (again, no matter how small), but I think we’ve had enough proofs for one day!)
First, thanks to Ian Tice and Ryan Rice for checking over my proofs and not throwing too much of a fuss over me playing somewhat fast and loose with the notions of equality and infinity. They are actual mathematicians, I just play one on myspace. So while some of the arguments I present here are not technically as simple as they seem, they should be close enough to get an idea of what is going on. Mathematicians may cringe, but the rest of us can rest easy and marvel at such things as the equality of the set of primes and natural numbers.
Second, if any of this at all interested you, I would definitely recommend David Foster Wallace’s Everything and More. I hear some mathematicians harumph at the somewhat conversational style and minor nitpicky errors he makes, but it is a fascinating book that really covers all the cool stuff about infinity with understandable proofs as to why it is so.
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