Writing is thinking. To write well is to think clearly. That's why it's so hard.

~ David McCullough

# Infinity Plus One

2014年09月21日

Everyone has, at some point during their childhood, found themselves in a contest to name the largest number. It starts off with perhaps a hundred, or a thousand, then a million, a billion, a trillion. Perhaps the savvy youngsters will know about quadrilions or quintillions.

But at some point, we run out of words for numbers. The argument continues with one-upsmanship. Your friend names a number. You tack "plus one" onto the end. They come back with "plus two". Maybe some multiplication gets thrown in there ("a trillion trillion!").

The game gets boring pretty quickly, though. You decide it's time to pull out your trump card: infinity!

Surely, you've won. No one can top infinity. It's bigger than the biggest number, after all. Your friend will have to forfeit.

Or so you think. But they come right back at you: "Infinity plus one!"

## The Aftermath

From that point on, the argument degenerates into whether or not infinity equals infinity plus one. Or perhaps what two times infinity is equal to. Regardless of "who won", neither side is particularly confident in their answer (although each is sure the other is dead wrong).

This game doesn't get much better after elementary school. A frustrated high schooler, unhappy to lose the argument will make the claim infinity isn't actually a number. And in college, they will take the logical next step and tell you that it's a "limit" instead and that you can't ever "get there".

But the argument never feels fully resolved.

## The Study of Infinity

There's a widely held belief that infinity is a mysterious and mystical concept. That no one really understands it. In older times, infinity was associated with the divine. How could a mere mortal comprehend something as boundless as infinity? And maybe there's some truth to that. But it's not the whole story.

Some of our greatest minds have thought about infinity for thousands of years. It's inevitable that we would have learned at least a little about it during that time. And indeed, modern mathematics has many, many different ways of looking at infinity and making sense out of it without getting caught in the traps we fell into back in school.

But the average person learns hardly anything at all about infinity in math class. Maybe you might see it in calculus, but for the most part, it's just something left to the students' imaginations. And since a high schooler doesn't have a thousand years to research it on his or her own, he or she is left with that feeling of mysticism.

Perhaps a pragmatic educator might try to justify this. Just as the concept of the number zero was considered abstract and useless to ancient people (Why would you say "\$0" when you could say "no money"?), infinity is the same way today. But while zero eventually found its way into the curriculum because of its use in science, engineering, and economics, infinity has not.

This shouldn't be too surprising. While zero appears smack dab in the middle of the number line, infinity appears at the distant ends of it. Most of the mathematics we are taught in school is meant to train us to keep track of the important numbers in the world. And the values that we deal with on a day-to-day basis are almost always finite. The places where infinities do crop up are dealt with sloppily in calculus classes. But this approach doesn't seem to cause any major issues overall.

But there is mathematics beyond algebra and calculus. And it is there we must go to understand the true nature of infinity.

## The Extended Real Numbers

The first place we notice a problem is in the seemingly innocent operation of adding one to infinity. Our intuition brings us to two contradicting results. The first is that infinity is the largest number there is. The second is that adding one to any number gives you an even larger number. There is also a strong intuition we have that says infinity ought to equal infinity plus one.

Let's step back for a moment. Let's think about what it means to add two numbers together. In the case that a and b are finite numbers, we certainly know how to add them. We learned how to do so in elementary school. We were never taught, however, what to do in the case that either of a or b (or both) are infinite.

Let me give you a quick lesson on mathematics. A little secret that is never taught in school. At least, not to non-math-majors. It's a lesson I learned from a professor of mine, Dr. El-Zanati, in my Linear Algebra class my sophomore year of college. And one I will never forget:

In mathematics, we are free to choose whatever definitions we like. But once we choose them, we must stick with them and suffer their consequences.

So, since your teacher didn't give you the definition, we are free to extend the definition of a + b to whatever we want when a or b is infinite. So let's follow our intuition, and give the following definition:

• If a and b are finite, then a + b is define as it usually is.
• If a is finite, then ∞ + a = ∞ = a + ∞.
• If a is finite, then -∞ + a = -∞ = a + -∞.
• ∞ + ∞ = ∞
• -∞ + -∞ = -∞
• Lastly, -∞ + ∞ = 0 = ∞ + -∞.

These definitions should hopefully be agreeable. One thing to notice is that I have also decided to include a "negative" infinity, -∞, and when you add -∞ to +∞, the two annihilate each other.

Now that we have our definitions straight (assuming we agree to adopt this definition), our argument is resolved. We see that ∞ + 1 = ∞. And infinity is, in fact, the largest number you can name.

But what does my teacher mean when he says 'suffer [the] consequences'?

Well it turns out, if we introduce infinity in this fashion, we end up breaking some of the basic rules of algebra! In particular, we lose associativity (the law that says parentheses don't matter in addition). Behold:

  (∞ + ∞) + -∞
=      ∞  + -∞
=      0


But if we rearrange the parentheses, we get:

  ∞ + (∞ + -∞)
= ∞ + 0
= ∞


So if algebra breaks down, does that mean infinity doesn't exist? Not necessarily. It just says that if infinity does exist, then some of the rules of algebra don't work. But if we throw the broken rules away, there is no problem, and we can promote infinity to the status of 'actual number'.

As quirky as this number system is, it is well-known and useful to mathematicains. We call these the extended real numbers.

## What is a number, really?

Numbers are something we're exposed to from the day we're born. They're so familiar to us that we think we know what one is when we see it. But I want to ask a somewhat rhetorical question: What is a number, really?

You might begin to answer this question by taking inventory of the kinds of numbers you already know about. You know the natural numbers: 0, 1, 2, 3, 4, 5, ... that are used to count things. You know about fractions, for when you invite your friends over for pie. When you get to algebra, your teacher starts yammering on about "real numbers". Later on, it makes a bit more sense when he or she explains there are also "imaginary" and after that, "complex" numbers.

But, at their heart, what are numbers about? Again, the question is rhetorical, and I wouldn't dare propose a singular, absolute answer. But speaking generally, numbers are defined, not one-by-one, but in collections. It will typically come with at least two operations, addition and multiplication, defined for every pair of numbers in it. (Although, what constitutes "addition" and "multiplication" is up for interpretation). Very often, there are other operations also defined on them: subtraction, division, exponentiation, roots, etc.

So as a first approximation, numbers are a collection of things which you can apply something vaguely addition-like and multiplication-like operations to. And perhaps there will be a few other operations as well. Because the numbers and operations are so interdependent, we sometimes use the phrase number system.

But there's more to it.

## Ordering for a number system

Many (but not all) number systems have a notion of ordering. We agree that 2 is larger than 1, but less than 3. And, by our argument above, infinity (as we defined it as part of the extended real line) is the greatest number of all.

But between different systems of numbers, the exact nature of the ordering can vary. Among the real numbers, there is no largest number: we can always add one to get a bigger number. (The same is true in the integers). The real numbers don't have a smallest number either. But the natural numbers do. (It's just 0). When we extend the real numbers with ∞ and -∞, we get both a smallest and a largest number.

For many applications, having a largest and smallest number is a handy property. All subsets are closed under two operations known as the infimum and the supremum, which act a lot like minimum and maximum respectively. These operations turn out to be important in the foundations of calculus. So if you were upset that we had to throw away some of our laws of algebra, perhaps you can be happy knowing that in doing so, we got something good out of it.

The ordering differs even between the real numbers and integers. Given any integer, you can always find the "next one up" by adding one. Perhaps frustratingly, there is no "next" number above a given real: between 0 and 1, there's 1/2. Between 0 and 1/2, there's 1/4. between 0 and 1/4, there's 1/8. And so on.

Some number systems don't have a suitable ordering at all. The complex numbers, for instance, rather than forming a 'one-dimensional' line, form a 'two-dimensional' plane. The idea of order doesn't make any sense, because there's no sensible way to compare, say, 1 and i. The number 1 appears further to the right in the plane while the number i appears further up.

## Topology for a number system

In the case of the complex numbers, instead of an ordering, we must work with a more subtle notion: topology. The full definition is is fairly technical, and I won't attempt to explain it here. But the gist of it is that a topology allows us to talk about which points are close to others. It tells us how a space is connected to itself. It is also used to give a precise definitions for curves and continuity.

One way to illustrating the meaning of a topology is this. Suppose you surgically removed the number 0 from the number line. You'd have no way to "walk" from -1 to +1. On the other hand, if you removed 0 from the complex plane, you could easily walk around the hole, taking a path from -1 to i, then from i to +1, without falling in.

The topology of the extended reals is fairly similar to that of the regular real numbers. The extended reals do possess a certain notable technical condition called compactness, but I will leave that to curious readers to research on their own.

## The Projective Line

Now that we have a broader view of what a number might be, let's look at some more ways to look at infinity.

In the extended reals, we have two infinities: +∞ and -∞. But what if we just had one? In fact, let's imagine that we took the extended line and "glued" the two infinities together. What would we get?

The resulting number system is called the projective line. Unlike the extended reals, which have a very good notion of ordering, the projective line throws that all away. But what we lose in one area, we gain in another. The projective line has a more interesting topology.

The reason is this. The very large positive numbers are now, in a sense, connected to the very negative numbers. If you were to poke a hole and remove the 0 again, you could still get from +1 to -1 by walking to infinity (indeed, a very far distnace to travel!), and then traveling past it. You would find yourself very far into the negative numbers, without having crossed 0. You have avoided the hole.

Geometrically, the projective line is easy to visualize as a circle. It's as if you took the number line, rolled in into a ball, then connected the two distant ends with a new point: the point at infinity.

## A Short Aside and Homogeneous Coordinates

Just to clarify, all of what I'm saying is grounded in the very precise language of mathematics. Don't get the impression that a mathematician's job is to close his eyes and imagine rolling lines into circles like a baker rolling dough. The playful imagery I'm using is typical in mathematics, but it must always be backed by a more formal analysis. Intuition is an incredible tour guide, but it is easily fooled in tricky situations.

While I want to keep this article fairly non-technical, I will give a brief explanation of how to represent the numbers of the projective line more formally.

In elementary school, we learn about fractions and how to manipulate them. In particular, we learn that a single fraction can be written multiple ways: 1/2 and 2/4 are really the same number. In effect, a fraction is just a pair of integers, subject to some rules for when they are considered equal.

The numbers of the projective line are very similar. It's just that the rule for comparing them is different. The traditional notation is to write a number as [a:b] where a and b are real numbers. This notation is known as homogeneous coordinates.

Just as a fraction is not allowed to have a zero denominator, for a number [a:b] on the projective line, at least one of a or b must be nonzero: [0:0] is undefined.

Two homogeneous coordinates [a:b] and [c:d] are considered equal if the components are in proportion to each other. That is, "a is to b as c is to d". Another way of saying this is that if r is any nonzero number, [a:b] is equal to [ra:rb]. So [1:2] and [2:4] are the same, and [2:0] and [4:0] are the same, while [1:2] and [1:3] are different.

Using homogeneous coordinates, we get a simple description for all numbers on the projective line. It turns out that all "finite" numbers can be written in the form [a:1]. The point at infinity, on the other hand, turns out to be the point [1:0]. If you imagine [a:b] as a fraction a/b, this looks curiously like division by zero leading to infinity, giving some justification for the name.

## The Riemann Sphere

Let's again engage our creative side. Both the extended real numbers and the projective line took the real number line and augmented it with either one or two infinities. What if instead of starting with the real numbers, we started with the complex numbers?

One such adaptation of the story of infinity was made famous by a mathematician named Riemann. (You may know his name from calculus class). In this number system, we start with all of the complex numbers, then add just a single point of infinity. How do you get to that point? Just travel outward from 0 along a straight line for an infinite distance. If you continue to travel the same direction passsed the point at infinity, you end up very far out in the opposite side of the origin (but still lying on the same line).

This number system is well-known to students who go on to take complex analysis. (Essentially, calculus over the complex numbers). It is interesting because, while not all the laws of algebra apply, you are now allowed to divide by 0! (Although, 0/0 is still left undefined). Because of this and its association with the complex numbers, it is especially good for studying rational functions, the name given to ratios of polynomials

Geometrically, you can think of the riemann sphere forming (surprise) a sphere. Again, you imagine curling the complex plane into a ball, then attaching the ends with the new point at infinity.

The riemann sphere is also known as the projective complex line, and when defined rigorously, the definition exactly parallels the projective (real) line in the last section.

## The Projective Plane

Here's one last variation on the "projective" theme. What if we took the plane and, instead of adding just one infinity, we add one infinity for every direction we could travel. The result is called the projective plane.

In the projective plane, if you started out going left from the origin, you would eventually wrap around and come out the right side. If you traveled up, you'd eventually wrap around to the bottom. But while in the riemann sphere, these two paths would cross at the (one and only) point at infinity, in the projective plane, they would travel across different points at infinity, and would not cross until you came back to the origin. In fact, every pair of straight lines in this space intersect at exactly one point -- even parallel lines.

The projecive plane is an exotic place. Geometrically, it can't be visualized exactly in 3D space. It has a special property of being non-orientable, which means if you took a trip passing any infinity, when you wrapped around, everyone would appear to be driving on the wrong side of the road. All your right-handed friends would suddenly be left-handed, and your shirts would be awkward to button down.

Nevertheless, this number system has found many appications in real life. In art, the projective plane explains the behavior of perspective and vanishing points. Perhaps you have heard of one-, two-, and three-point perspetive in your art classes. One way of looking at perspective is that there are always three vanishing points in a scene. It's just that sometimes they end up "at infinity". (An astute reader might wonder, "Which infinity? There are many." To which the answer is, "Look for the parallel lines in the scene").

The numbers that make up the projective plane can be described with homogeneous coordinates with three components instead of two: [a:b:c]. The projective plane itself might also be viewed as gluing a projective line to the ends of a regular plane.

I can't resist making one last provacative claim. If we got just one more dimension, we get to "projective 3-space" (describable with homogeneous coordinates with four components). While this space is even crazier and harder to imagine than the projective plane, it turns out, again, to have real importance in the world. It explains how rotations work in 3D space, and in particular, its non-orientability can be used to explain the physical phenomenon of gimbal lock.

You may have noticed the last few examples slowly drifted away from numbers and into geometry. This is actually fairly typical in modern mathematics. Numbers (and algebra) have an intimate relationship with geometry. Numbers are very precise tools, but as any highschooler knows, are hard to work with. Geometry is less precise, but is far easier to intuit. A great deal of mathematics revolves around translating between the two worlds: we use geometry to figure out what ought to be true, and we use numbers (and algebra) to prove it rigorously.

But enough about space. Let's look at some completely different ways of thinking about infinity.

## The two problems of calculus

If one sees infinity at all during his or her high school or college curriculum, it will usually be in calculus class. The two central problems in calculus are finding tangent lines to curves and calculating the area under curves.

Let's consider the first problem: finding the tangent line to a curve at a point. A good strategy seems to be to pick a point near the one you're interested in and draw the line that passes through both points. This is called a secant line. If the second point is near the first, it will give a good approximation to the tangent. If you choose an even closer point, it will give an even better approximation. How do you find the tangent line exactly?

Naively, you might think to move the two points so that they are on top of each other. But alas, you now only have one point, yet it takes two points to define a line. Looking at the problem through the lens of algebra, we see that the slope of the line is given by the change in y over change in x. But when the two points coincide, the slope would have been given as 0/0. Clearly nonsense.

Now, let's look at the second problem of finding the area under a curve. (More technically, it's signed area, but nevermind that). We might get an approximation by fitting rectangles underneath the curve. If we make the widths of our rectangles small, we get a good approximation. If we shrink the size of our rectangles (adding more rectangles in the process to fill in the rest), we get a more accurate approximation. But again, naively, the best approximation of all seems to require us to work with an infinite number of zero-width rectangles. Again, nonsense!

So we are faced with two problems: division of zero by zero and summing an infinite number of infinitesimal quantities.

## Sequences and Limits

Both of these problems end up being solved using the same tool: the limit. Like many useful tools in math, the precise definition of a limit is a bit technical. (Although for the curious, it is very closely related to the notion of topology we talked about earlier). The intuition, however, is very clear after a little prep work.

A sequence is an infinite list of numbers given by some rule. Here are some examples:

• 0, 1, 2, 3, 4, ..., the sequence of natural numbers
• 1, 1/2, 1/4, 1/8, 1/16, 1/32, ..., the sequence of reciprocal powers of 2
• 3, 3.1, 3.14, 3.141, 3.1415, ..., the sequence of the digits of π
• 1, -1, 1, -1, 1, -1, ..., the sequence of alternating signs of 1.

Some sequences seem to trend towards a value. In the reciprocal powers of 2, each number in the sequence gets smaller and smaller, quickly approaching 0. The digits of π, of course, approach the number π. In these cases, we say the sequence converges towards those respective values, and the value approached are called the limit of the sequence.

Not all sequences have limits, though. The other two examples above do not trend toward any (finite) number. They are said to diverge. By some definitions of limit, we might work in the extended real line, and in that case, we would actually say the first converges at +∞. The last example never settles, though, and never has a limit.

Try to come up with various examples of sequences on your own. Which converge? Which don't? See if you can come up with an example where you can't tell right away whether it converges or diverges.

Armed with our basic knowledge of limits, we can tackle the problems of calculus successfully. In the case of the tangent line, we consider the sequence of slopes of successively more accurate secant lines. The slope of the tangent line will be the limit of this sequence.

In exactly the same way, when calculating the area under a curve, we construct a sequence of the successively more accuarate approximations by rectangles under the curve. Again, the limit of this sequence turns out to be the actual area.

Calculus, together with its big brother analysis, might be sumemd up as the study of ways clever people have learned to apply limits to things. If we sum progressively longer chains of numbers, we get infinite series. A taylor series is a way to define an infinite-degree polynomial. They allow us to calculate sine, cosine, the exponential function, and the logarithm function with arbitrary precision. The fourier series allow us to break down periodic functions into their frequency components. It was absolutely instramental to the viability of radio and broadcast television, and it is the basis of modern digital audio and video compression.

## A bit of set theory

So far, we've only mentioned various ways that infinity crops up in settings that have a very "continuous" feel to them. We have not said anything yet about infinity as a number of things that you can have.

The standard foundation of modern mathematics since the 1890s has been Cantor's set theory. A set is a collection of mathematical objects -- often numbers, but also more abstract objects such as points, lines, functions, graphs, or even other sets.

To give a few examples, I might talk about the set of natural numbers. Or the set {1, 2, 3}. Or maybe the set of prime numbers. Or the set of all natural numbers less than a billion.

Some sets contain only finitely many things. Others are infinite. The set of three digit numbers is finite. But the set of numbers with more than three digits is infinite, as is the set of real numbers.

Finite sets and infinite sets are very different beasts. If I'm given enough paper and ink, I should be able to (in principle) list out every element of a finite set. Equivalently, if I were to remove items from a finite set one-by-one, I would eventually exhaust it, leaving me with an empty collection. Neither of these two things is possible with an infinite set. If I remove a billion elements from an infinite set, the remainding set will still be infinite.

## Cardinality

Cardinality is the name given to the measure of the size of a set in set theory. For finite sets, this is simply the number of elements in the set.

Given two finite sets, we might want to know if they have the same number of elements. One way to find out is to remove elements from each set, one from one, one from the other, over and over, until one or both sets is exhausted. If one is empty before the other, we have discovered one of the sets to have more elements.

A natural question to ask would be: Can we compare the size of infinite sets in any meaningful way?

To answer this, you might look to the examples you know well: the integers. You notice that the integers form an infinite set. You also notice that the set of even integers also forms an infinite set. Because the even integers make up a (strict) subset of all the integers, there must be fewer.

But there's a subtle caveat to this thinking.

To see it, consider which set is greater in size. The set of even integers? Or the set of odd integers?

You might argue either way. The integers alternate fairly between even, odd, even, odd, so perhaps they are equal in number.

But, if you cut them into positives and negatives, you could imagine the positive evens would cancel out with the positive odds and the negative evens would cancel out with the negative odds.... and zero (and even number) would be left standing. So many there's one more even number.

But what's so special about zero? We could make the cut at 1 instead. A similar argument would apply, and we would find there's an extra odd number.

So which is it?

To make the problem even more apparent, what if we took the rational real numbers and compared them to the irrational real numbers. It's not at all clear how we would compare them in the first place!

## Hilbert's Grand Hotel

The mistake here is that which elements are included in the set should not have any bearing on its size. If we take a set (such as the odd integers above) and rename every one of them (perhaps replacing each odd number n with the fraction n/2), the cardinality should not be affected.

More investigation shows us that this renaming of elements might be a good principle to work with. Cantor noticed this and used it as a definition for when any two sets (infinite or not) are the same size. In particular, he said two sets are equal in cardinality if each element of the first can be paired off with an element of the second.

With finite sets, this pairing is trivial to produce: just choose any arbitrary pair of elements (one from each set) one-by-one until you run out. But with infinite sets, it might require some cleverness to pair them off. And critically, to show that two infinite sets are of different cardinalities, it's not enough to simply fail to pair them. You must show that no pairing could ever be possible.

The surprising result is that, under this definition, we can pair the integers and the even integers using the following rule: send n in the integers to 2n in the even integers (and, going in reverse, send each even integer m to the integer m/2).

Each integer is now paired with an even integer, and vice versa. Every number in both sets is accounted for. And we must conclude that the cardinality of the integers and the even integers is the same.

There is an entertaining story named after the famous mathematician Hilbert about a hotel with an infinite number of rooms, one for each natural number. The hotel is full, but then a new guest shows up. Can the hotel find a room for him? The story has been told many times by many others much better than I could, and it's definitely a treat to hear.

## Even larger infinities

With this definition for when two infinite sets are equal in cardinality, you might be curious to ask: Are all infinite sets the same size?

The answer, perhaps surprisingly, turns out to be no! There is a trick to prove that for every set, there is another set whose elements are too numerous to be paired off, no matter how hard you try.

The most commonly-cited example is that there are "more" real numbers than there are integers. The proof works basically the same, but I prefer a slightly different version: there are more sequences of natural numbers than there are natural numbers themselves.

The proof is one by contradiction. You assume that I, an undoubtedly clever person, claims to have come up with a pairing. For each natural number n, I have a matching sequence s(n).

You, an even cleverer person, are dubious of my claim. You think about it for a while, and you decide to refute my claim (or any other similar one I make later) by showing me a sequence I have forgotten to include. That is, you will produce a sequence which is different from each sequence in my list.

Remember that you may define a sequence by giving a rule for which elements appear in it. The rule you choose is this: the first element of your sequence is s(1) + 1. (Remember s(1) is the sequence I chose to pair up with the number 1). The second element of your sequence is s(2) + 1. The third element is s(3) + 1. And so on.

You hand me this sequence and ask me to find which number n it corresponds to. But I run into trouble. Your sequence differs from s(1) because the first entry in your sequence is one greater. Thus, it can't correspond to 1. It also differs from s(2) because the second entry in yours is one greater, so it can't correspond to 2. And similarly, I find that your sequence will always differ from the nth sequence on my list because between them, the nth entries will always differ by 1.

In the end, I must admit to you that I lied. My pairing did not include every sequence as advertised, because it couldn't. Indeed, no pairing ever could.

This proof technique is often called Cantor's diagonal argument. The name makes sense if you imagine the infinite list of sequences written out in a table. After I present you the list, you procede by looking at the entries along the diagonal and generating your counterexample by adding one to each as you pass it.

The same technique shows up in computer science. It is closely related to the halting problem.

## The Set of all sets

One last abuse of infinity I feel is worth pointing out is the notion of a set so large that it contains all sets. Such a set would be enormous, containing all the finite sets, all the integers, all the reals, and even itself!

Again, though, paradox strikes. If a set that large did exist, there would be another set (using an argument similar to the one above) that must be even larger. But if a larger set exists, we would be 'missing' some sets from the smaller set of sets.

Set theorists worked very hard during the early 1900s trying to show that paradoxes did not crop up in set theory. They wanted to use it as a foundation for all of mathematics. To this end, the set of all sets was abolished. In the rare instances that mathematicians wanted to talk about it, they were urged to use the term class instead. A class behaves in almost all regards as a set, but 'regular' sets may not contain classes, and it is perfectly admissible to include a class of all sets.

However, in the 1960s, a new branch of mathematics became popular, especially for the study of algebra and geometry. Category theory, as it is known, studies very large collections that are sometimes too large for even classes to capture.

One solution to these ever growing "size issues" was proposed by Alexandre Grothendieck, a man who almost single-handedly revolutionized a field known as algebraic geometry. His work paved way for Andrew Wiles to prove the famous Fermat Theorem in 1994. His personal and political life is also fascinating: at the end of his incredibly successful career, he gave up mathematics and retreated from civilization. No one today knows what became of him.

A grothendieck universe is a set which behaves just like a set of all sets ought to. While it does not contain every set in a literal sense, it is closed under all the operations sets ought to. If were were restricted to working just with the sets included inside a grothendieck universe, there would be no set you would feel you were missing. Yet, you can work with them freely, without worry about running into such a paradox.

Grothendieck universes are powerful enough that you cannot prove they exist in the standard set theory. Instead, it is typical to include one or more of them as axioms. Assuming their existence leads to no known contradictions.

## Ordinals

To the brave souls who have made it this far with the journey, I will leave you with one more tangible example of how infinity can be used in mathematics.

Whereas the cardinal numbers tell you how many elements are in a set, ordinal numbers tell you where each element stands. They are how you rank elements in a set so that you always know "who's up next".

Imagine the elements of your set are people lined up in front of a restaurant, waiting to be seated. Each takes a numbered ticket and waits for their number to be called. The numbers we use are called ordinals, but what do they look like?

For finite sets, ordinals are just natural numbers. Being trained as a computer scientist, I will unapologetically start counting with 0 instead of 1. The first patron in line gets 0. The second gets 1. The third gets 2. The fourth gets 3. And so on.

But as I hope you have picked up by now, the transition from finite to infinite can be a bumpy ride.

Suppose we went to a restaurant where women received priority seating. And furthermore, let's assume that the restaurant received such high recommendations than an infinite number of people came to line up, evenly split between men and women.

The first person in line, a woman, received the number 0. The second person in line, another woman, received the number 1. The third (again a woman), receives 2. For miles, the line consists of only women, each holding a ticket with a number on it. It stretches on forever.

Yet, after all the women in line, we get to all the men in line. They are standing in an orderly fashion, too. But what kind of number do we assign the first man? The exact construction is, again, technical. But we'll just say that the first infinite ordinal is traditionaly named ω, so the first man is assigned a ticket labeled as such. The second man in line gets ω + 1 and the third, ω + 2. And so down the line the men stretch for the rest of the block.

It may seem unfair to the men. Even the first man in line has an infinite wait ahead of him. If the restaurant seats guests one at a time, surely no man will ever be seated.

But it's still worthwhile for the men to keep track of their places regardless.

With the line being as long as it is, even a very large finite numbered ticket will result in a long wait. If a patron has to wait too long in line, he or she may become impatient and leave. And so while it may start off that the man with ticket ω has an infinite wait time, if all but a finite number of women get tired of waiting and leave, he will find himself with a long -- but no longer infinite -- wait time before being seated.

## The Wellordering Principle

Ordinals mathematically have to do with a principle known as wellordering of sets. A wellordered set behaves in many respect like the natural numbers (which are the prototypical example of such a set). In particular, a wellordered set is susceptible to a proof technique called transfinite induction, where you "iterate" over an infinite set, making sure to hit every element, even if some elements are "infinitely distant" in the order.

It is sometimes assumed as an axiom in set theory that every set has a wellordering. This axiom is known to be equal in strength to a number of other statements, the most famous of which are the Axiom of Choice and Zorn's Lemma. Historically, this trio caused a lot of philosophical headache for mathematicians. They simultaneously allow you to prove many useful statements alongside many seemingly impossible ones. For instance, the wellordering axiom can be used to prove that it is possible to cut up a ball into finitely many pieces, rotate and move those pieces around, and end up with two copies of the ball you started with, each exactly the same size. Yet, if we abolish this axiom, we must also admit the strange possibility that you might take the cartesean product of a bunch of nonempty sets and end up with a set that is empty.

The question of the admissibility of the wellordering principle truly is a testament to my professor's remark about suffering the consequences of your definitions.

## Wrangling infinity

All successful attempts to work with infinity in mathematics involve an indirect attack. As we learned as children, there is no hope of success if you start at 0` and count one-by-one up to infinity forever. You'll never actually reach the end.

Instead, we look for clever ways to come up with a finite means to crack the problem. We find finite descriptions of the infinite things we're interested in.

In the case of the extended real line and the projective plane, we reduced infinity to a mere number with clearly-defined rules for working with it. In calculus, we reduce infinite sums to limits of finite sums. While I omitted the details, a limit itself is a lot like finding a winning strategy (a fixed set of guidelines) for a game. With cardinalities, we do not consider the items of a set one at a time, but rather, we find a rule to pair them all up at once.

In some of the cases not discussed here, we look for a finite value associated with the infinite object in quesiton. These tend to go by names like degree, dimension, or rank. In cases where the infinite objects are not tame enough to get the results we want, we may restrict some aspect while letting the rest remain infinite. Examples of this are notherian modules in algebra, compact spaces in analysis, or CW complexes in algebraic topology.

By identifying the finiteness buried at the heart of the unbound infinite, we are able to approach our problems with the best understood, most versatile, and ultimately simplest mathematical tool we have at our disposal: the act of counting.

I hope I have helped you to better appreciate the many ways mathematicians have interpreted infinity. What I have talked about here is hardly an exhaustive list, but a curiousity and a willingness to explore mathematics will take you as far as you want.

qed