You probably have some idea of what *infinity* means. Something that is infinite goes on forever. But it might surprise you to know that there are different kinds of infinity, and some are bigger than others!

As a simple example, a small circle is infinite in the sense that you can loop around and around the circle forever. At the same time, your entire path along the circle is *bounded* in the small area of the circle. Compare that to the straight line that extends to infinity. If you travel that line, you follow a path that goes forever in some direction.

What if we draw a larger circle outside the small circle. If there are an infinite number of points on the small circle and an infinite number of points on the large circle, does the larger circle have the same number of points as the small one? [The answer is *yes*.]

To understand all this, we have to first talk a bit about numbers.

We can start with the *counting* numbers. They have been with us the longest. From the beginning humans have found it useful to count things: sheep, coins, people; even ships and mountains.

Before numbers were really invented, humans used a one-to-one relationship between what was being counted and some way of representing the count. Hash marks in clay, pebbles in a bag, knots in a rope, are all such representations.

If you’ve ever counted something using the “prisoner’s hash marks,” you’ve used this early counting method.

Technically, it’s called base one notation. One mark (or knot or pebble) stands for one *something*. A rope with 30 knots is a lot easier to cart around than 30 sheep. Or 30 ships!

Because the process of counting things is natural, the counting numbers are formally known as the *natural numbers*.

Mathematicians have a symbol that represents the natural number; the symbol is **N** (or **ℕ**).

Many of the natural numbers have names: *one*, *two*, *three*,… *fifteen*, *fifty*, *hundred*, and so on. Others are named by tacking names together: *fifty-seven*, *twelve-hundred-and-ninety-nine*, and so on.

It might surprise you, but there is no agreement on whether *zero* is included in the natural numbers. Some do; some don’t.

(In part, this is due to zero having come along later in western mathematics. At first there was no symbol to stand for the unmarked clay, the unknotted rope, the empty pebble bag.)

The next step along the way was the need to sometimes represent a *part* of something.

If one day’s work was worth one loaf of bread; a half day’s work is worth a half loaf. How many loaves does a worker who worked 10 half days get?

Somehow you had to add up 10 half loaves.

That example is simple enough, but it’s just the start.

Once you develop a way to handle *fractions* of things, it gets harder. In school, you’ve probably had to add **2/3** to **1/5**, and that’s not as intuitive as adding ten halves!

Numbers which are fractions, that can be represented as **a/b**, are called *rational numbers*; their symbol is **Q** (or **ℚ**), for *quotient*.

All natural numbers are also rational numbers. For instance, five is **5/1**.

Rational numbers have a *decimal expansion* form. For example, **1/2** is **0.5**, **1/5** is **0.2** and **1/3** is **0.333…**

That last one illustrates an important point: the decimal form of rational numbers extends forever. In fact, it *always* extends forever. Technically speaking, **1/2** is **0.500…**, and **1/5** is **0.2000…**

Another important case comes from **1/11**, which is **0.090909…**

The decimal expansion of rational numbers either repeats the same digit or it repeats a pattern of digits.

A more complex pattern comes from **2/7**, which gives us **0.285714285714285714…**

If there are rational numbers, you might think there could be *irrational* numbers.

You’d be right! All numbers with a decimal point are called *real numbers*; they have the symbol **R** (or **ℝ**). Real numbers with repeating sequences are *rational*; real numbers that never repeat are *irrational*.

*Irrational numbers* cannot be represented with fractions (because all fractions result in rational numbers).

Nearly *all* real numbers are irrational numbers! The most famous irrational number is π (pi). The digits of π go on forever and never repeat.

Now we’re finally ready to get back to the idea of infinity.

You probably know that the natural numbers, the counting numbers, are infinite.

You can start counting and keep counting forever. No matter how big a number you name, you can always add a one to it to make a bigger number.

This fact is actually the theoretical basis of the natural numbers.

The process of adding a one to a number to get the next number in line is called a *successor function*.

We can count the natural numbers *because* there is a successor function (the successor function called “add one”).

Which brings us to the first important concept of infinity: *There is a type of infinity that can be counted.*

The counting goes on forever, but you can always name the next number in line, no matter where you are in the count.

It turns out that even the rational numbers are countable.

You can design a successor function that enumerates the rational numbers, because rational numbers have the form **a/b**, and since ‘**a**‘ and ‘**b**‘ are natural numbers (which are countable), rational numbers are countable.

This brings us to the second important concept of infinity: *All countable infinities are the same “size.”*

Suppose you list the natural numbers: 1, 2, 3,…

And then you also list them by tens: 10, 20, 30,…

Even though the second list skips nine numbers for every number on the list, both lists are the same size! (The same logic applies if we take, for example, just the even numbers: 2, 4, 6…)

We prove this by showing that, for every number on the first list, there is a corresponding number on the second list (and vice-versa).

We call this *mapping* one *set* (list) onto another. If the map is one-to-one, then the set size must be the same.

The set of natural numbers and the set of rational numbers are also the same size.

You might think the set of fractions was bigger with all those fractions between each natural number, but the two sets can be mapped onto each other, which means they must be the same size.

The same size of infinity. *Countable* infinity.

Now consider the irrational numbers.

What is the first number after π?

We know the digits of π go on forever with no pattern. A successor function adds something to the current number to generate a new number. What can we add to π to generate the next number? We need something along the lines of an infinity of zeros followed by a one.

But infinities go on forever, so we never stop with the zeros, which means we never get to the one!

And that means we can *never name* the next number after π. Or the one right below it.

In fact, for any given real number (natural, rational or irrational), we can *never* name the next irrational number.

Which brings us to the third important concept about infinity: *The irrational numbers cannot be counted, because there is no successor function.*

What this ends up meaning is that the set of all irrational numbers is a *larger* infinity than the set of natural (or rational) numbers. There is no map between the two sets.

About 130 years ago, a mathematician by the name of Georg Cantor came up with an elegant proof demonstrating that the real numbers (which includes rationals and irrationals) were uncountable.

His proof uses a trick, called the diagonal argument, that I’ll explain when we take this topic up again.

For now, remember these facts about infinity:

- The natural and rational numbers are infinite but countable.
- All countable infinities are the same size.
- The irrational numbers are infinite and uncountable.
- Uncountable infinities have a larger “size” than countable infinities.

Until next time, don’t count your chicken enchiladas until they’re served!

Update 11/10/2014: The diagonal argument is discussed in detail in the Sideband #54: Cantor’s Diagonal post.

October 20th, 2012 at 4:51 pm

I had to read this about 3 times before firmly grasping the concepts. I imagine it would take an infinite amount of time for me to truly comprehend. I love math, but my brain has decided to chuck it in favor of knowing all the lyrics to top hits from the 80s. C’est la vie.

October 20th, 2012 at 5:04 pm

Awesome that you did read it three times to get the comprehension, though!!

Thing about those 80s lyrics over math… you’d be the one winning the trivia contests. They don’t have many questions about the theory of integers. (Pity. Maybe I’d do well then. I really, really suck at trivia. I forget most books and movies not long after reading/seeing them… it does make it nice to repeat them, though.)

October 22nd, 2012 at 12:18 pm

Wow. I loved the part about cheese enchiladas! YUM!

Seriously I’m not completely daft. I watch a lot of TV and one thing I know about Infinity I learned from an astronaut and a very rich lady with a black belt and a plan for Revenge

1) Infinity and Beyond!

2) Infinity x infinity as a measure of love.

Google it! *head nodding vigorously* See I am smart!! 🙂

October 22nd, 2012 at 2:54 pm

Indeed! What’s more, Infinity x Enchiladas =

Super Yummy!But…

…does it have Enchiladas inside?

October 23rd, 2012 at 10:56 am

Nice! In fact, I used to begin ALL my math classes (Pre-Algebra and beyond) with a lecture just like this! One question: I did not think that the rationals were countable (discrete), since there can always be another in between any two of them.

October 23rd, 2012 at 9:49 pm

I know! Astonishing isn’t it! And yet they are. Both the numerator and the denominator are integers, so they have to be countable. The diagram with the caption, “Count on the blue line,” shows how it’s done. That back and forth pattern continues to infinity, but the path covers every possible rational.

Here’s a problem you might like chewing on (or maybe it’ll be child’s play for you). Several times now, on theoretical math blogs, I’ve seen someone present a trick they claim enumerates the reals. Basically, it’s a tree; it can be binary or decimal or whatever. Let’s say it’s decimal. Top node represents the decimal point. The ten nodes beneath it represent the real numbers 0.0 – 0.9. Each node has ten nodes. Now the tree goes from 0.00 – 0.99. The tree extends downwards to infinity.

Therefore, every possible real number is in the tree (that part is true). Therefore the tree can be enumerated, which means Cantor was wrong, right? (Wrong! But explaining why is trickier than it seems, which is why people think they’ve proved Cantor wrong. (At least it took some figuring for me; as I said, maybe it’s obvious to you.))

October 26th, 2012 at 11:29 pm

Interesting stuff. Can’t claim much of it stuck, but infinity is awesome to ponder. 🙂

October 27th, 2012 at 10:44 am

Infinitely so! 😀

April 17th, 2015 at 7:55 pm

So next time someone says “I’m infinity times infinity cooler than you.” I can reply, “Well I’m irrational infinity!”

(Of course, that person could reply: “I’m irrational infinity times irrational infinity.”)

April 18th, 2015 at 2:01 pm

Ha! Yes, exactly! You can go full math geek on them and point out that “infinity times infinity” is still the same plain old infinity, and they haven’t actually increased

anything. XDNow you’ll understand and appreciate the phrase, “Don’t worry if math makes no sense to you — most numbers are

irrational!” (There’s a tee-shirt that reads: “Numbers may be real, but most are irrational.”)Funny thing… doing math with infinities is something mathematicians actually do. It’s called transfinite math. What little I know about it gives me the same sense of the willies that complex (imaginary) numbers do to you. That sense of, “No… you just made

thatshit up outta the whole cloth!” I wonder if there isn’t a point like that in mathematics for many — a point where it just gets too weird.As you know, there is a division among mathematicians between whether math is just a game of symbol manipulation (with no fundamental connection to reality) or whether it’s somehow a reflection of the real world. It’s kind of a modern math philosophy view of the Plato/Aristotle divide. (I’ve heard it said that most working mathematicians are secret Platonists because otherwise they’re devoting their lives to a made up game. These days, given the popularity of video games, perhaps that’s not as embarrassing seeming as it once might have been.)

Anyway, there seems a point for many where the mind rebels and goes, “Oh, no!

Thatis just a game with symbols!” For me it’s the idea of a “population” of infinities. I can see the countable and uncountable division. To me that mirrors the digital and analog division we see throughout reality. (If matter and energy are quantized (discrete) — which we know they are — but time and space are smooth (per Einstein, but we don’t know if it’s true), then the “bumpy” and “smooth” division exists in the very fabric of reality.)April 18th, 2015 at 8:53 pm

The real/game divide in math is interesting. I can see how the secret Platonists might just say, “Well THAT part’s a game, but the rest is real,” each drawing their lines according to what they think makes sense. I have no idea what happens with real closet Platonist mathematicians, but I’d be curious to know.

April 19th, 2015 at 12:32 pm

I’m not quite sure what you’re asking in your question… Do you mean what they do in the privacy of their own closets? ❓

There are a variety of points of view in mathematical philosophy. Formalism, Intuitionalism, Constructivism, and Fictionalism, are all different ways of looking at (and seriously questioning) math’s reality.

As you probably know, I lean towards Realism, even a form of Platonism. I do think think that math is

a priori. At the same time, while I’m not sure I’m entirely clear on Kant’s analytical-synthetic distinction, I agree with his view that math isa priorisynthetic (insofar as I understand “synthetic”). I agree thereisan element of human synthesis. Wediscovermathematical principles and objects, but weinventthe language and meta-structure of mathematics based on those discoveries.A crude metaphor: dogs are real objects we discover, but every language has different words for them, and every society sees, places, and uses, them within that society differently.

April 19th, 2015 at 12:54 pm

Well you know I’m not a math person. I’m not sure where I am on mathematical philosophy, but of course I’d like to be a Platonist. 🙂

So what do “closet Platonist” mathematicians make of the really bizarre math like imaginary numbers? Are imaginary numbers in some sense “real” for them? My question was, do these mathematical Platonists have to disregard some useful math as “not real” or merely manipulation while holding that most of it is real?

BTW, I found this article you might like. No need to read it if you don’t feel like it. I skimmed it and decided it was over my head, but it might not be over yours.

http://www.friesian.com/imagine.htm

April 19th, 2015 at 2:36 pm

Ha! I have some pages about Kant I printed off that site many years ago. I’ve been meaning to go back and see if the site was still around and do some more reading, so thanks for the link!

I’m no expert, but I’d

guessPlatonists would accept the reality of the complex number plane as a real form. Did Plato require a form be expressible in concrete form in our world? Or can an inexpressible perfect form merely haveapplicationin our world?The Intuitionalists (and some others) would be the ones most likely to choke on complex numbers, I’d think.

April 19th, 2015 at 7:40 pm

I feel like I should know the answers to those questions, but I don’t. The details about the forms are often obscure, and so is their relationship to the visible world. Scholars debate whether Plato really bought into his theory of forms since he actually criticized the theory himself.

Glad you liked the link!

April 19th, 2015 at 9:33 pm

It was funny you gave me that link. I’m not kidding that I’ve been meaning to see if that site still exists for a long time. The stuff about Kant was really instructive. I re-read it occasionally over breakfast as I chew on trying to understand Kant. But it sometimes makes reference to parts I didn’t print, so — this is over breakfast mind you — I keep reminding myself to type in the link printed along the bottom of the page and see if the site is still around.

But being me, when I’m actually at the computer, I never remember. Now I have an actual link I can ignore just like the one to the Standford Encyclopedia of Philosophy that I liked so much I donated a bit of money and have always meant to spent a lot of time reading. For years. [sigh]

April 20th, 2015 at 5:58 pm

Wow, that’s some pretty dense reading over breakfast!

April 21st, 2015 at 2:33 pm

I’ve always liked reading while eating breakfast. It started with cereal boxes, of course, the breakfast-reading gateway. That lead to bringing my own reading material, but breakfast is short enough to make fiction kinda pointless, so it’s invariably something non-fiction.

June 7th, 2022 at 2:00 pm

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