Is infinity plus one bigger than infinity?
What is infinity?
In a sense we all have an inkling of what infinity is. It’s something that characterises things that never end. The list of natural numbers 1, 2, 3, 4, . for example, or an infinitely long straight line that starts at a point just in front of you and stretches on forever, straight ahead. You can’t reach the end of that line even if you travel in the fastest car, and no matter how long you count for, you can never reach the end of the natural numbers.
But are these two types of infinity, that of the natural numbers and that of the line, the same? Intuitively you might think that they are different. The natural numbers are separate «things», while the line is all connected up. You could place the natural numbers along your line, at distance 1 metre apart. This gives a sense that there is somehow more to the infinity of the line than to the infinity of the natural numbers: the line is able to fill the gaps between the numbers.
Mathematicians agree with that intuition. They distinguish between countable infinities and uncountable ones. The natural numbers form a countable infinity, and that makes sense, as you could count through all of them if you had an infinite amount of time. A group of infinitely many people also qualifies as a countable infinity. That’s because (with an infinite amount of time) you could make a list of all the names, each name taking its own place on the list, and then you could count through them, just as you can count through the natural numbers. In general, an infinite collection of objects forms a countable infinity if you can list the objects one by one, with a place on the list for every object and one object for every place on the list.
What about the infinitely long straight line? It is also made up of infinitely many objects: in this case the objects are points on the line. If you imagine this line as an infinitely long ruler, then each point comes with a number: the starting point of the line comes with the number 0, the point half a metre along comes with the number 0.5, and so on (the collection of numbers you get from the ruler are called the positive real numbers). Can you make a list of those numbers to show that they also form a countable infinity?
One approach would be to order those numbers by size. But that quickly gets you into trouble. Clearly the first number should be 0, but what about the second one? You could try 0.1, but then, 0.01 is smaller than that, so it should come before 0.1. But what about 0.001? For every number you might designate as taking the second place on the list you can find a smaller one (you simply insert an extra 0 after the decimal point). So listing those numbers along the ruler by size is hopeless.
Could there be another way of listing them? The answer is no. There is a mathematical argument (see here) which shows that any list of positive real numbers definitely misses out at least one other positive real number. You can never make a complete list. This shows that the infinity represented by the infinite straight line (or, equivalently, the positive real numbers) is an uncountable infinity.
Which infinity is bigger?
What about the idea that the infinity of the infinite line is somehow «bigger» than the infinity of the natural numbers? One way of comparing the size of finite collections of things, if you can’t be bothered to count, is to see if you can match them up exactly. Think of a number of chairs and a number of people. If there is a chair for every person and no chairs are left over, then you know that there must be the same number of chairs as people. If there are chairs left over, then you know that there are more chairs than people. And if there are some people left standing, you know that there are more people than chairs.
You can extend this idea to infinite collections of objects. If you can match the objects in collection A to the objects in collection B exactly, with every object in A corresponding to exactly one object in B and vice versa, then we say that the two collection have the same size or, as mathematicians put it, the same cardinality. We have already seen this in action with our infinite group of people above. By listing them one by one we have actually matched them up exactly with the natural numbers: for every person there is exactly one natural number (their place on the list) and for every natural number there is exactly one person (the one occupying the place on the list given by that natural number). This is why we say that the group of people and the natural numbers represent the same kind of infinity — a countable one.
Going back to points on the infinite line, however, it turns out that any attempt to list them (to match them exactly with the natural numbers) misses out at least one point. This is why we say that the cardinality of the line (an uncountable infinity) is greater than the cardinality of the natural numbers (a countable infinity).
Intuitively, uncountable infinities appear more unwieldy and tricky than countable ones and in mathematics they often are. But this doesn’t mean that countable ones are straight-forward. As an example, think of all the even numbers 2, 4, 6, 8, and so on. There are infinitely many, but what is the cardinality («size») of that infinity compared to that of all the natural numbers? Surely it should be half as big?
The answer is no. We said that two infinite collections have the same cardinality if the objects in one can be matched up exactly with the object in the other. It’s pretty easy to match all the even numbers exactly with all the natural numbers:
So the cardinality of the even numbers is the same as that of the natural ones. If this seems weird then perhaps the next result is even weirder. It’s possible to show (see here) that all the rational numbers (that is, all the fractions such as 1/2 or 5/6) can also be listed; that is, they can be matched exactly with the natural numbers. So even though there appear to be many more fractions than natural numbers (there are infinitely many fractions between any two successive natural numbers) the two sets of numbers have the same cardinality.
The eminent Galileo Galilei discovered such strange facts about infinity back in the 17th century and thought they were so weird, it put him off thinking about infinity claiming that «we cannot speak of infinite quantities as being the one greater or less than or equal to another». Over 200 years later, the mathematician Georg Cantor picked those ideas up again, undeterred by their weirdness, and went much further. He discovered a whole tower of infinities, each one larger than the other. The infinities of the natural numbers and the infinity of the line are only two of them.
About this article
A version of this article first appeared on the Plus website.
Some Infinities Are Bigger Than Others But There’s No Biggest One
But we can show that the number of points on the interval zero to one is a bigger infinity than the counting numbers are. The first clue is the fact that we can’t count the number of points on a line interval.
Try labeling the points on a line as points 1, 2, 3, etc. No matter what scheme you come up with, there will always be some points on the line segment that are not included in your count.
Georg Cantor (1845–1918) came up with an ingenious argument to show that the infinite number of points on a line was greater than the number of counting numbers. His method, called diagonalization, is illustrated in the figure shown:
Recall that the infinities of two sets are the same if there is a one to one correspondence between the two sets. The number of even numbers and the number of primes were both shown to be the same set size as the set of counting numbers. There is a one-to-one correspondence between the two sets.
If the points on a line were the same infinite size as the set of counting numbers, then we should find such a one-to-one correspondence. This is what the table shown attempts. The number 0.84935296… is being assigned to the first counting number 1. Remember, this point on the line is irrational and can go on forever. For the counting number 2, we assign the point 0.96540047… The table goes on and assigns a different number on the line interval to every counting number. Can there be a one-to-one correspondence?
Look at the number X at the bottom of the table. This number does not appear on the list. The first digit in X, an 8, cannot be the first number on the list whose first digit is 9. It can’t be the second number on the list either. The second number has the number 6 as its second digit and the second number in X is a 7. The same applies to the number assigned to the counting number 3. Its third digit is a 0 and the third digit in X is a 1. On and on we go. The number X can’t be the fourth number, the fifth, and on and on to infinity. The number X is not on the list of numbers.
So the number of points on a line segment is greater than the number of counting numbers. It is a bigger infinity.
Cantor called the countable infinite, the infinity of all counting numbers, ℵ0. ℵ is the Hebrew letter aleph. The number of points on a line segment, a bigger infinity, is denoted by ℵ1. We are immediately prompted to ask if there is an even bigger infinity? The answer is yes. ℵ2 can be thought of as all of the set of points, squiggles, clumps of points, and combinations thereof that can be written in a square.
Cantor showed that a bigger infinity can always be constructed by taking the set of all subsets of a lower infinity. In general, there are 2 M subsets of a set with M elements. If there are 3 elements in a set, there should be 2 3 =2×2×2=8 subsets. The eight subsets are A, B, C, AB, AC, BC, ABC and the null set. So a higher infinity than ℵn is the set of all subsets of ℵn . ℵn+1 is equal to 2 raised to the power of ℵn.
So there is no biggest infinity! A larger infinity can always be constructed.
Interestingly, I know of no one who can visualize ℵ3 and beyond. We have the math, but what does it mean? An analogy is our perception of 3 spatial dimensions. If I stretch my mind, I can visualize 4 spatial dimensions. But outside of a mathematical description, I cannot grasp the physical interpretation of 5 or more spatial dimensions.
We started off Part 1 of this series on infinities with an example of the lowly shepherd who resorted to stone counting to keep track of his sheep. Counting beyond 4 or 5 was simply beyond the shepherd’s ability so he gathered stones to make a one-to-one correspondence. We might scoff, but here we are seen to have similar limitations. Like the shepherd, we must resort to math ideas to deal with infinities above ℵ2 and spatial dimensions above 4. This is beyond our limited ability. But not beyond God’s. No wonder Georg Cantor sought an audience with Pope Leo XIII to discuss the theological implications of his theory of the infinite.
So ends our discussion of the infinite. It is a beautiful theory but infinity does not exist in reality. If it did, we’d have a lot of ludicrous and counterintuitive consequences to deal with.
We hope you enjoy this series on the unique, reality-defying nature of mathematical infinities. Here are all five parts — and a bonus:
Part 1: Why infinity does not exist in reality. A few examples will show the absurd results that come from assuming that infinity exists in the world around us as it does in math. In a series of five posts, I explain the difference between what infinity means — and doesn’t mean — as a concept.
Part 2. Infinity illustrates that the universe has a beginning. The logical consequences of a literally infinite past are absurd, as a simple illustration will show. The absurdities that an infinite past time would create, while not a definitive mathematical proof, are solid evidence that our universe had a beginning.
Part 3. In infinity, lines and squares have an equal number of points Robert J. Marks: We can demonstrate this fact with simple diagram. This counterintuitive result, driven by Cantor’s theory of infinities is strange. Nevertheless, it is a valid property of the infinite.
Part 4. How almost any numbers can encode the Library of Congress. Robert J. Marks: That’s a weird, counterintuitive — but quite real — consequence of the concept of infinity in math. Math: Almost every number between zero and one, randomly chosen by coin flipping, will at some point contain the binary encoding of the Library of Congress.
Part 5: Some infinities are bigger than others but there’s no biggest one Georg Cantor came up with an ingenious proof that infinities can differ in size even though both remain infinite. In this short five-part series, we show that infinity is a beautiful — and provable — theory in math that can’t exist in reality without ludicrous consequences.
You may also wish to read: Yes, you can manipulate infinity in math. The hyperreals are bigger (and smaller) than your average number — and better! (Jonathan Bartlett)
Robert J. Marks II
Director, Senior Fellow, Walter Bradley Center for Natural & Artificial Intelligence Robert J. Marks Ph.D. is Senior Fellow and Director of the Bradley Center and is Distinguished Professor of Electrical and Computer Engineering at Baylor University. Marks is a Fellow of both the Institute of Electrical and Electronic Engineers (IEEE) and Optica (formerly the Optical Society of America). He was the former Editor-in-Chief of the IEEE Transactions on Neural Networks and is the current Editor-in-Chief of BIO-Complexity. Marks is author of the books Non-Computable You: What You Do That Artificial Intelligence Never Will Never Do and The Case For Killer Robots. He is co-author of the books For a Greater Purpose: The Life and Legacy of Walter Bradley, Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks and Introduction to Evolutionary Informatics. For more information, see Dr. Marks’s expanded bio. Follow Bob Profile