When I was younger, my teacher made a curious statement: “There are as many even numbers as there are whole numbers.” At first, this seemed puzzling. How could this be true when even numbers are just a subset of all numbers? Let’s delve into this intriguing concept of infinity and see what it really means.
To grasp the idea of infinity, we need to understand what it means for two sets to be the same size. Imagine comparing the fingers on your two hands. You don’t need to count them to know they’re equal; you can simply match each finger on one hand with a finger on the other. This concept of one-to-one matching is crucial in understanding set sizes.
Historically, some cultures used this method to keep track of quantities. For example, shepherds would use stones to represent sheep leaving a pen, ensuring each sheep had a corresponding stone. This way, they could tell if any sheep were missing without actually counting them.
Similarly, if you see a full auditorium with no empty seats, you know there are as many people as there are chairs, even if you don’t know the exact number. This is the essence of matching sets.
Returning to the idea of even numbers and whole numbers, consider listing all whole numbers and pairing each with its double. This creates a perfect one-to-one match between whole numbers and even numbers, showing they are the same size in terms of infinity.
Despite the intuitive feeling that even numbers are just a part of whole numbers, this matching demonstrates that both sets are infinite and equal in size.
Now, let’s consider fractions. Can we list all possible fractions? It seems daunting, but Georg Cantor devised a clever method in the late 1800s. He arranged fractions in a grid and created a list by sweeping diagonally, skipping duplicates. This method establishes a one-to-one match between whole numbers and fractions, suggesting they are also the same size in terms of infinity.
However, not all numbers are fractions. Numbers like the square root of two and pi are irrational, meaning they cannot be expressed as simple fractions. These numbers are represented by infinite, non-repeating decimals.
Can we list all decimal numbers, including both rational and irrational numbers? Cantor showed that it’s impossible. If you claim to have a complete list, a new decimal can always be constructed that isn’t on your list, proving the list incomplete. This demonstrates that the set of decimal numbers is a larger infinity than the set of whole numbers.
This leads to the astounding conclusion that there are different sizes of infinity. Cantor also showed that for any infinite set, creating a new set of all its subsets results in a larger infinity. This means there are infinitely many infinities, each larger than the last.
These ideas were initially controversial, even causing Cantor personal distress. However, they are now fundamental in mathematics, accepted by researchers and taught to students worldwide.
Cantor’s exploration of infinity led to the continuum hypothesis, questioning whether there are infinities between the size of whole numbers and real numbers. In the 20th century, Kurt Gödel and Paul J. Cohen showed that this hypothesis can neither be proven true nor false, revealing that some mathematical questions are unanswerable.
This discovery highlights the limitations of mathematics, yet it also showcases the fascinating complexity of the field. Infinity, with its many layers and mysteries, continues to captivate and challenge our understanding.
Pair up with a classmate and use objects like coins or cards to practice one-to-one matching. Try to create a one-to-one correspondence between two sets of objects, such as matching coins to cards. Discuss how this exercise helps you understand the concept of infinite sets being the same size.
Engage in a debate with your peers about the statement “There are as many even numbers as there are whole numbers.” Use arguments from the article to support your stance. This will help you articulate and deepen your understanding of infinite set sizes.
Work in small groups to explore Cantor’s diagonal argument. Create a list of fractions and attempt to construct a new number that isn’t on your list. Discuss how this exercise demonstrates the concept of different sizes of infinity.
Research and present on an irrational number, such as pi or the square root of two. Explain why these numbers cannot be expressed as fractions and how they fit into the concept of infinity. This will enhance your understanding of irrational numbers and their place in mathematics.
Participate in a class discussion about the continuum hypothesis and its implications. Consider why some mathematical questions are unanswerable and what this means for the field of mathematics. This activity will encourage critical thinking about the limitations and possibilities within mathematics.
When I was in fourth grade, my teacher said to us one day: “There are as many even numbers as there are numbers.” I thought, “Really?” Well, yes, there are infinitely many of both, so I suppose there are the same number of them. But even numbers are only part of the whole numbers; all the odd numbers are left over, so there must be more whole numbers than even numbers, right?
To understand what my teacher was getting at, let’s first think about what it means for two sets to be the same size. What do I mean when I say I have the same number of fingers on my right hand as I do on my left hand? Of course, I have five fingers on each, but it’s actually simpler than that. I don’t have to count; I only need to see that I can match them up one to one.
In fact, some ancient people who spoke languages that didn’t have words for numbers greater than three used this sort of matching. For instance, if you let your sheep out of a pen to graze, you can keep track of how many went out by setting aside a stone for each one and putting those stones back one by one when the sheep return, so you know if any are missing without really counting.
As another example of matching being more fundamental than counting, if I’m speaking to a packed auditorium where every seat is taken and no one is standing, I know that there are the same number of chairs as people in the audience, even though I don’t know how many there are of either.
So, what we really mean when we say that two sets are the same size is that the elements in those sets can be matched up one by one in some way. My fourth-grade teacher showed us the whole numbers laid out in a row, and below each, we have its double. As you can see, the bottom row contains all the even numbers, and we have a one-to-one match. That is, there are as many even numbers as there are numbers.
But what still bothers us is our distress over the fact that even numbers seem to be only part of the whole numbers. But does this convince you that I don’t have the same number of fingers on my right hand as I do on my left? Of course not. It doesn’t matter if you try to match the elements in some way and it doesn’t work; that doesn’t convince us of anything. If you can find one way in which the elements of two sets do match up, then we say those two sets have the same number of elements.
Can you make a list of all the fractions? This might be hard; there are a lot of fractions! And it’s not obvious what to put first or how to be sure all of them are on the list. Nevertheless, there is a very clever way that we can make a list of all the fractions. This was first done by Georg Cantor in the late 1800s.
First, we put all the fractions into a grid. They’re all there. For instance, you can find 117/243 in the 117th row and 243rd column. Now we make a list out of this by starting at the upper left and sweeping back and forth diagonally, skipping over any fraction that represents the same number as one we’ve already picked. We get a list of all the fractions, which means we’ve created a one-to-one match between the whole numbers and the fractions, despite the fact that we thought maybe there ought to be more fractions.
Now, here’s where it gets really interesting. You may know that not all real numbers— that is, not all the numbers on a number line— are fractions. The square root of two and pi, for instance. Any number like this is called irrational. Not because it’s crazy, but because the fractions are ratios of whole numbers and are called rationals; meaning the rest are non-rational, that is, irrational. Irrationals are represented by infinite, non-repeating decimals.
So, can we make a one-to-one match between the whole numbers and the set of all the decimals, both the rationals and the irrationals? That is, can we make a list of all the decimal numbers? Cantor showed that you can’t. Not merely that we don’t know how, but that it can’t be done.
Look, suppose you claim you have made a list of all the decimals. I’m going to show you that you didn’t succeed by producing a decimal that is not on your list. I’ll construct my decimal one place at a time. For the first decimal place of my number, I’ll look at the first decimal place of your first number. If it’s a one, I’ll make mine a two; otherwise, I’ll make mine a one. For the second place of my number, I’ll look at the second place of your second number. Again, if yours is a one, I’ll make mine a two, and otherwise, I’ll make mine a one.
See how this is going? The decimal I’ve produced can’t be on your list. Why? Could it be, say, your 143rd number? No, because the 143rd place of my decimal is different from the 143rd place of your 143rd number. I made it that way. Your list is incomplete. It doesn’t contain my decimal number. And, no matter what list you give me, I can do the same thing and produce a decimal that’s not on that list.
So we’re faced with this astounding conclusion: The decimal numbers cannot be put on a list. They represent a bigger infinity than the infinity of whole numbers. So, even though we’re familiar with only a few irrationals, like the square root of two and pi, the infinity of irrationals is actually greater than the infinity of fractions.
Someone once said that the rationals—the fractions—are like the stars in the night sky. The irrationals are like the blackness. Cantor also showed that for any infinite set, forming a new set made of all the subsets of the original set represents a bigger infinity than that original set. This means that once you have one infinity, you can always make a bigger one by making the set of all subsets of that first set. And then an even bigger one by making the set of all the subsets of that one. And so on.
And so, there are an infinite number of infinities of different sizes. If these ideas make you uncomfortable, you are not alone. Some of the greatest mathematicians of Cantor’s day were very upset with this. They tried to make these different infinities irrelevant, to make mathematics work without them somehow. Cantor was even vilified personally, and it got so bad for him that he suffered severe depression and spent the last half of his life in and out of mental institutions.
But eventually, his ideas won out. Today, they’re considered fundamental and magnificent. All research mathematicians accept these ideas, every college math major learns them, and I’ve explained them to you in a few minutes. Someday, perhaps, they’ll be common knowledge.
There’s more. We just pointed out that the set of decimal numbers—that is, the real numbers—is a bigger infinity than the set of whole numbers. Cantor wondered whether there are infinities of different sizes between these two infinities. He didn’t believe there were, but couldn’t prove it. Cantor’s conjecture became known as the continuum hypothesis. In 1900, the great mathematician David Hilbert listed the continuum hypothesis as the most important unsolved problem in mathematics.
The 20th century saw a resolution of this problem, but in a completely unexpected, paradigm-shattering way. In the 1920s, Kurt Gödel showed that you can never prove that the continuum hypothesis is false. Then, in the 1960s, Paul J. Cohen showed that you can never prove that the continuum hypothesis is true. Taken together, these results mean that there are unanswerable questions in mathematics. A very stunning conclusion. Mathematics is rightly considered the pinnacle of human reasoning, but we now know that even mathematics has its limitations. Still, mathematics has some truly amazing things for us to think about.
Infinity – A concept in mathematics that describes something without any bound or larger than any number. – In calculus, the limit of a function as it approaches infinity can help determine its end behavior.
Numbers – Mathematical objects used to count, measure, and label. – Real numbers include both rational and irrational numbers, providing a complete set for analysis.
Sets – A collection of distinct objects, considered as an object in its own right. – The union of two sets A and B is a set containing all elements from both A and B.
Matching – A correspondence between two sets where each element of one set is paired with an element of the other set. – In graph theory, a matching is a set of edges without common vertices.
Fractions – Mathematical expressions representing the division of one integer by another. – Simplifying fractions is a fundamental skill in algebra to solve equations efficiently.
Irrational – Numbers that cannot be expressed as a simple fraction, having non-repeating, non-terminating decimal expansions. – The number π is an example of an irrational number, crucial in geometry and trigonometry.
Decimals – Numbers expressed in the base-10 numeral system, which includes a decimal point to represent fractions. – Converting fractions to decimals is a common task in algebra to facilitate easier computation.
Sizes – The magnitude or dimensions of a mathematical object, often referring to the number of elements in a set. – The size of a matrix is determined by its number of rows and columns.
Hypothesis – An assumption or proposition made for the sake of argument, often used as a starting point for further investigation. – In mathematical proofs, a hypothesis is the initial statement that is assumed to be true for the purpose of proving a theorem.
Mathematics – The abstract science of number, quantity, and space, either as abstract concepts or as applied to other disciplines. – Mathematics is essential for developing models that predict real-world phenomena in physics and engineering.
