What is Infinity?

One of the common errors made by calculus learners is to consider infinity as a number. True, there are number systems such as the extended real number system in which infinity is successfully treated as a number, and you can ponder on them if you wish, but for our purposes at this level of learning calculus, infinity is decidedly not a number.

Numbers satisfy various properties, and infinity does not satisfy these properties. For example, it might seem reasonable to state that $\infty + 1 = \infty$ (how could this be any different?), but then by the usual properties of numbers, we should be able to subtract $\infty$ from both sides of the equation to obtain $1 = 0$, which is nonsense. This is enough to rule out infinity as a number, but you can have fun deriving all kinds of contradictions based on the assumption that it is a number, for your own amusement. (It won’t take you long to prove that all real numbers are equal, starting from the crazy assumption that $1 = 0$, for example, which is further reinforcement that the assumption that infinity is a number is not valid.)

Be alert to the use of the symbol $\infty$ in various arguments in calculus textbooks for various purposes. Recognize that it is a time-saving symbol that represents various facts or processes. While you are striving to understand the facts and processes it represents, remind yourself regularly that infinity, while a very useful concept, is not a number.

So what is infinity, then?

Let’s start by thinking about the natural numbers. A basic property of the natural numbers is that you can always add the number $1$ to a natural number, and the result is another natural number. For example, $7$ is a natural number, and it is possible to add $1$ to $7$, with the result being $8$, another natural number. Now this is true no matter how large the natural number you choose, because this is a property of all natural numbers. What, then is the largest natural number?

You will be able to understand that based on this property of natural numbers, there is no largest natural number. Suppose someone proposes to you that some natural number, no matter how large, is the largest natural number. You could counter this proposal by simply adding $1$ to the proposed largest natural number to produce a natural number that is even larger. But that new natural number is not the largest either, because you can also add $1$ to it to obtain an even larger one.

So there is no largest natural number. One could say that there is an unlimited number of natural numbers. Another way to say this is that there is an infinite number of natural numbers. This usage of the word infinite summarizes the fact that there is an unlimited number of natural numbers.

There are many cars on Earth, but if you had to do so, you could count all of them. You could take a super-snapshot of Earth at a particular time, and then you could carefully examine this photograph and count all of the cars on Earth. No doubt the number is very large, but it is a natural number. The number of cars on Earth is finite, because in principle you could count the number and the result is a natural number. Similarly, you could (in principle) count all of the atoms on Earth, at a particular time, and this too is a natural number, so we say the number of atoms on Earth is finite.1

Similarly, there are collections of numbers that are finite. For example, consider the collection of odd numbers that are between $1$ and $100$ inclusive. There are $50$ such numbers, right? We could say that the set of odd natural numbers less than or equal to $100$ is finite. You could easily construct any number of finite sets, such as the set of prime natural numbers that are less than $1000$, the set of even numbers between $5000$ and $9000$ inclusive, and so on. So we have finite sets, and then we have infinite sets, such as the set of all natural numbers.

In order to be able to have some way of talking about the number of elements of a set in a unified way, whether the set is finite or infinite, mathematicians have coined the term cardinality. It doesn’t make sense to speak about the number of elements in the set of all natural numbers, because there is no such number. It does make sense to speak about the number of elements in the set of odd numbers between $1$ and $100$ inclusive; this number is $50$. We can say that the cardinality of the set described in the previous sentence is $50$, and the cardinality of the set of all natural numbers is infinite. Thus, the concept of cardinality gives us a way of speaking about the “size” of a set, whether the number is finite or infinite.

Introducing the concept of cardinality may seem unnecessary, but let’s discuss something that is potentially shocking. It certainly shocked numerous mathematicians when they learned about it from Georg Cantor about a century ago:

There are different infinities, of different “sizes.” If you prefer, there are different “levels” of infinity.

Is this not mind-boggling?? Are there really infinite sets that have different cardinalities?

To understand this amazing fact about infinities (yes, we should use the plural now), we’ll first have to think about how to compare the cardinalities of two sets that are infinite. For example, imagine the set of all natural numbers (we’ll call it $A$), and then imagine the set that includes all natural numbers and that also includes the number $0$ as well; we’ll call this set $B$. Now it seems reasonable to say that set $B$ is larger than set $A$; after all, set $B$ includes everything in set $A$, and set $B$ also includes one number that is not in $A$. In the language of set theory, we would say that set $A$ is a proper subset of set $B$. However, this is not the way we currently understand infinite sets, as you will see in the next few paragraphs.

Cantor came up with a criterion for comparing infinite sets that led him to his revolutionary understanding of infinities. He said that two sets have the same cardinality if you could set up a one-to-one correspondence between the two sets. That is, you have to be able to pair the elements of the two sets, so that each pairing matches an element of one set with an element of the other set, no element of either set is included in more than one pairing, and each element of each set is included in some pairing.

Think about a sports stadium with $50,000$ seats. Now imagine that the stadium is full of people, so that each seat is occupied by one person, no seats are empty, and each person in the stadium is in a seat. You don’t need to count the people to determine how many of them are in the stadium; you can immediately conclude that there are $50,000$ people in the stadium, because they are in one-to-one correspondence with the seats, and you know how many seats there are.

Now let’s apply this concept of comparing cardinalities to infinite sets. In particular, consider the sets $A$ and $B$ described a few paragraphs ago. There exists a one-to-one correspondence between the sets $A$ and $B$, and so they have the same cardinality! Even though we have some sort of sense that we should be considering $B$ to be bigger than $A$, this sense is not applicable to infinite sets. According to Cantor’s definition of equal cardinalities, these two sets have the same cardinality! Can you come up with a formula for a one-to-one correspondence that confirms this?

A good way to intuitively understand this is through the story of Hilbert’s Hotel. David Hilbert was one of the greatest mathematicians of about a century ago, and he constructed a series of “thought experiments” involving a hypothetical hotel that has an infinite number of hotel rooms. I imagine the hotel rooms all in a (very long!) row, rather like a motel, to model the natural numbers as we would normally plot them along a number line.

Suppose that all of the infinite number of rooms in Hilbert’s Hotel are occupied at the moment, so that there are no vacant rooms. If a new prospective guests walks into the hotel’s lobby desperately asking for a room for the night, is he or she out of luck? Well, not necessarily, according to the clever front desk clerk. The clerk merely asks each guest in the hotel to vacate their room and shift one room over. That is, the person in Room 1 moves to Room 2, the person in Room 2 moves to Room 3, and so on. Each existing guest is perfectly well-accommodated in their new room, but now Room 1 has been vacated, and so it is available for the new guest. Problem solved!

Once you have let this remarkable solution sink in, you might then understand why there is a temptation among some people to express this shifty business as
$$\infty + 1 = \infty$$
Resist the temptation to do this! This kind of equation encourages us to treat $\infty$ as if it were a number, but we have already argued that this is not so! So avoid such nonsensical equations. Nevertheless, you can also understand the temptation to write such an equation, because it does (in a way) capture an important property of Hilbert’s Hotel; even if all of the infinite number of rooms is occupied, space can always be made available for one more guest. Mind-boggling! Infinity sure is unusual.

Does this story help you to feel a bit better about the fact that the sets $A$ and $B$, described earlier, can be placed in one-to-one correspondence, and therefore have the same cardinality? Would it help more if you could find an explicit formula for such a one-to-one correspondence? Here’s one, perhaps the simplest one, that does the trick: $f(n) = n + 1$. The same formula describes the way existing guests must shift rooms: The guest in Room $n$ must shift to Room $n + 1$.

You can iterate the front desk clerk’s shifty technique to accommodate two new guests, three new guests, and indeed, any finite number of new guests. (What is the shifting formula in such cases if there are $m$ new guests?) But this kind of shifting clearly won’t work if an infinite number of new guests arrive, right? For example, let’s suppose that there is a neighbouring hotel that is much like Hilbert’s Hotel, in that there are an infinite number of rooms, all currently occupied by guests. There is a power outage at the other hotel; is there any way that this infinite number of guests can be squeezed into Hilbert’s Hotel, with each guest having his or her own room? If there were only $5$ new guests, we could just shift each existing guest five rooms over. But with an infinite number of new guests, shifting in this way doesn’t work. What does it mean to shift each existing guest an infinite number of rooms over? This is meaningless! Where does the guest currently in Room $37$ get moved? Because $\infty$ is not a number, saying that the guest in Room $37$ should be moved to Room $37 + \infty$ has no meaning!

But the front desk clerk is very clever, and decides that if for each $n$, the guest in Room $n$ shifts to Room $2n$, then each existing guest will still be accommodated (in all the even-numbered rooms), and yet an infinite number of rooms (the odd-numbered ones) will have been vacated, allowing all of the guests displaced from the other hotel to be accommodated also! Isn’t this amazing?

The previous paragraph shows that the cardinality of the even natural numbers is the same as the cardinality of the natural numbers. In some intuitive way, we would wish to say that there are half as many even numbers as natural numbers, but no, this intuition is not helpful when it comes to infinite sets. Similarly, the cardinality of the odd numbers is also the same as the cardinality of the natural numbers. What is a simple formula for a one-to-one correspondence demonstrating this latest fact?

Once again, one might be tempted to write
$$\infty + \infty = \infty \quad \quad \textrm{or} \quad \quad 2\infty = \infty$$
to express this strange fact, but one should really avoid doing so, as $\infty$ is not a number, and therefore can’t be combined in an equation like this according to the usual rules for manipulating numbers. But you can certainly see why such nonsensical equations are written in some places; they are attempts to express strange and wonderful properties of infinity in a form that is not appropriate for communicating such facts.

What if there were two or three other copies of the Hilbert Hotel, whose occupants all had to be squeezed into the Hilbert Hotel? Would you be able to do so if you were the desk clerk? Which formula proves that such redistributions of guests are possible? What if there were $m$ total copies of the Hilbert Hotel (including the HH); can you do the redistribution? What is a formula that proves that such a redistribution is possible?

If you were able to complete the tasks in the previous paragraph, you will now be convinced that the cardinality of $m$ copies of the natural numbers, taken as one giant set, is the same as the cardinality of one copy of the natural numbers by itself. Remarkable!

In the previous paragraph, $m$ is a finite number. What if you had an infinite number of hotels like the Hilbert Hotel? Would you be able to fit all of the guests in all of these infinite number of hotels into just one Hilbert Hotel by redistributing all of the guests? Surely this is impossible, right? At least it’s not possible using the method of the previous paragraphs for a finite number of copies of the natural numbers. It’s worth pausing right now, turning away, and mulling this over for some time. Return to your reading only after you have mulled things over for a while, and after having writing your thoughts in your research notebook.

After mulling it over, what do you think? In fact, it is indeed possible! The cardinality of an infinite number of copies of the natural numbers is the same as the cardinality of the natural numbers! Wow! It is a little more challenging to come up with an explicit formula for a one-to-one correspondence in this case. It may help you to sketch a diagram, where each row of the diagram corresponds to a copy of the natural numbers. Then ask yourself if there is a systematic way to step your way through the entire (infinite) array of numbers, such that you are certain to eventually step on each number in each row. Doing this may help you to understand that this is possible, and provided your pathway is simple enough, you may also be able to write a formula for the correspondence. This is a challenging task, but have fun with it!

We stated earlier on that there were different levels of infinity, but so far we have only encountered one, the cardinality of the natural numbers. Each of the infinite sets we have constructed so far has the same cardinality. It turns out that the cardinality of the real numbers is greater than the cardinality of the natural numbers. The proof that this is so is due to Cantor, again, and it is based on a beautiful idea nowadays called Cantor’s diagonal argument, which I’ll now describe.

Consider the real numbers between $0$ and $1$. Cantor showed that the cardinality of this set is not equal to the cardinality of the natural numbers by proving that it is not possible to place the two sets into one-to-one correspondence. He did this by using a proof by contradiction, which is to assume that it is possible and then demonstrate a contradiction, showing that the original assumption is false. So, let’s retrace Cantor’s steps by assuming that it is possible to construct a one-to-one correspondence between the natural numbers and the set of real numbers between $0$ and $1$. In effect, this assumption is that you can place the entire set of real numbers between $0$ and $1$ in a list in some way. For example, here is a partial list:

0.3715682…
0.4931657…
0.1153267…
0.0474749…
0.9535360…
0.0088841…
0.5583322

Clearly we can’t display the entire list, nor can we even show the complete decimal expansion of each number in the list, but the assumption is that this can be done. Cantor then argued that this assumption is incorrect by constructing a number that is not in the list. You can do this by constructing a number that differs from the first number in the first decimal place, differs from the second number in the second decimal place, differs from the third number in the third decimal place, and so on. You can do this according to some rule to make it easier; for example, if the given digit is a $3$, then make it a $5$, and if the digit is not a $3$, then make it a $3$. Look at the list of numbers above, and apply this rule to the red digits to construct a new number:
$$0.5333533…$$
The particular rule used is not essential; many other rules would work just as well. Consider the new number just constructed and note that it is not in the original list of numbers. You can tell it is not in the original list, because it is not the first number in the list (it differs in the first decimal digit), it is not the second number in the list (it differs in the second decimal digit), it is not the $47$-th number in the list (it differs in the $47$-th digit), and so on. Therefore, it is not in the list, and the assumption that we had a complete list of all real numbers between $0$ and $1$ is false.

Can you obtain a complete list of all real numbers between $0$ and $1$ by just including this new number at the top of the list? No. You can see that this attempt will not work by applying Cantor’s diagonal argument again to the new list to construct yet another real number between $0$ and $1$ that is not in the new list either. No matter how many newly constructed numbers you add to the top of the list, it will never be a complete list of all real numbers between $0$ and $1$.

The same argument can be applied to any proposed complete list of real numbers whatsoever. Isn’t this an ingenious argument? And isn’t the result absolutely remarkable?

Thus, it is not possible to list all of the real numbers between $0$ and $1$. Another way to say this is that it is not possible to place the real numbers between $0$ and $1$ in one-to-one correspondence with the natural numbers, and therefore the cardinality of the real numbers between $0$ and $1$ is different from the cardinality of the natural numbers.

It turns out that the cardinality of the set of all real numbers is the same (!!) as the cardinality of the set of real numbers between $0$ and $1$. Can you argue that this must be true? Hint: If you can construct a one-to-one function that maps the entire real line into the interval of real numbers from $0$ to $1$, then this would be an explicit proof. A function that maps the other way would work just as well. Search your memory banks for a graph from high school that will do the trick!

Isn’t it mind-boggling that the number of real numbers between $0$ and $1$ is the same (in the sense of one-to-one correspondence) as the number of all real numbers? These two sets have the same cardinality. Because the set of natural numbers is contained within the set of real numbers, and these two sets cannot be placed in one-to-one correspondence, we say that the cardinality of the real numbers is greater than the cardinality of the natural numbers. Thus, we have established the existence of two levels of infinity. Here is some standard terminology: Sets that either contain a finite number of elements or can be placed in one-to-one correspondence with the natural numbers (such as the even numbers, the odd numbers, the integers,2 and so on) are called countable sets. Infinite sets that are countable are also called countably infinite. Infinite sets that are not countable are called uncountable. Thus, we have (so far) two levels of infinite sets, sets that are countably infinite (such as the natural numbers) and sets that are uncountable (such as the real numbers).

Are there any levels of infinity that are between the cardinality of the natural numbers and the cardinality of the real numbers? Cantor conjectured in 1878 that the answer to this question is no, in what is now called the continuum hypothesis. Attempts were made for many years to either prove the continuum hypothesis or to discover a counterexample, which culminated in a publication by Kurt Gödel in 1940 in which he showed that it is impossible to disprove the continuum hypothesis within standard set theory. Paul Cohen showed in 1963 that the continuum hypothesis cannot be proved within standard set theory either! This remarkable set of results shows that the continuum hypothesis is independent from standard set theory. To learn more about this very strange result, look up Gödel’s incompleteness theorem.

We stated earlier that there is a whole hierarchy of infinities, but so far we have only seen examples of two levels of infinity, that of the natural numbers and that of the real numbers. How can one construct higher levels of infinity? What about the cardinality of the set of points in the plane that you used so much in high school to study functions? Surely the cardinality of the number of points in the plane is greater than the cardinality of the real line? But no, the cardinalities are the same! Proving this is more challenging, though, than Cantor’s diagonal argument. (Look up the Schröder-Bernstein theorem if you are curious about this.) Similarly, the cardinality of the points in three-dimensional space is also the same as the cardinality of the real number line. Thus, simply moving to higher-dimensional spaces does not give us a greater level of infinity.

How does one construct sets with cardinalities at higher levels of infinity? We shall leave this discussion for another time, but if you are curious you can consult a work on mathematical analysis or set theory.

Before concluding this discussion, it is worth mentioning that the ancient Greeks already distinguished between what they called actual infinity and potential infinity, and this is a useful distinctions. Actual infinity is reserved to describe an infinite set in its entirety, such as the set of natural numbers taken as a whole, or the set of real numbers taken as a whole. Potential infinity is reserved for the idea of a quantity that is increasing without bound, so that the quantity gets larger and larger with each step of the process, with no limitations on how large it gets. Our discussion of limits as $x$ “approaches infinity” fits this sense of potential infinity. In fact, we discuss limits as
$$x \rightarrow \infty \qquad \textrm{and} \qquad x \rightarrow -\infty$$
where we typically envision $x$ “moving” to the right indefinitely in the first case, and “moving” to the left indefinitely in the second case. It’s worth emphasizing again that in both of these cases “infinity” is not a place, but rather this is a process of imagining what happens when a quantity ($x$ in this case) either “moves” to the right indefinitely or “moves” to the left indefinitely.

1. It appears that some of the most serious problems we have on Earth is that we humans collectively treat some of our limited resources as if they were infinite instead of finite.

2. Can you prove that the integers can be placed in one-to-one correspondence with the natural numbers by constructing a suitable formula?