Spoiler! The answer's not infinity plus one. Heck, it's not even infinity times infinity. (Yes, I'm sad to say that ad with the guy in the suit sitting with the kids is lying to you.)
But there are ways infinity can be bigger than infinity. And actually, we can explain this idea using logic those kids sitting with that guy would understand. So buckle up, here are five big ideas you need to understand to wow your friends with epic math skillz at that next dinner party:
I. Set Theory:
Before kids learn to count, they learn to group stuff. Mathematician Steve Strogatz, in his excellent series on numbers for The New York Times, recalled a sketch from "Sesame Street" where the slow-witted Humphrey takes an order of fish by trying not to count them: "Go tell the kitchen fish, fish, fish, fish, fish fish," Humphrey tells an overwhelmed assistant. Mercifully, Ernie interjects, telling Humphrey it's a lot easier to just count the fish and tell the kitchen a number. Humphrey can't contain his amazement as he realizes how "this counting thing can really save a person a lot of trouble."
Essentially, what Humphrey was doing was set theory. He took a group of stuff and put it into a defined set: six fish. As Humphrey is excited to discover, that set could just have easily been a set of numbers {0, 1, 2, 3, 4, 5, 6 ... }, or a set of spark plugs, or cinnamon buns.
Humphrey learns it doesn't matter what specific items are in a set. All that matters is that there is stuff in the set. And that's set theory. Now that you've mastered that idea - you're probably wondering: how does this help me understand infinity?
II. Correspondence:
Correspondence is a concept in set theory that works by relating each individual item in one set to an individual item in another set. Those items could be things like Humphrey's six fish or they could be numbers {1, 2, 3, 4, 5, 6 ...}. With correspondence, the abstractions don't matter - all that matters is that each item in one set can be compared to an item in another set. For example, one fish to '1,' two fish to '2,' three fish to '3,' etc ...
In theory, Humphrey's set of fish could go on forever. This is called "cardinality" and it leads to the smallest type of infinity, which we like to call ...
III. Welcome to "Aleph-Null"
Before we go on, I have to stop for a second and say this post is heavily indebted to Alasdair Wilkins' outstanding essay, "A Brief Introduction to Infinity."
Let's get back to the math. Our question: how can one type of infinity be smaller than another? To understand that, let's take a basic example using two different sets of "counting" numbers (the fancy term for these numbers is "natural") and imagine those sets both extend forever. (I know there's some debate about zero being natural, but I'm a radio producer who studied history in college, so cut me some slack and let's just say it is.)
SET A: {0, 1, 2, 3, 4, 5 ...}
SET B: {1, 2, 3 , 4, 5, 6, ... }
As you can see, even though SET B begins one number higher than SET A, both sets contain the same amount of "stuff," which means they represent equal types of infinities. Said another way, every number in SET A could be corresponded to another number in SET B that is one value higher {0 corresponds to 1, 1 to 2, 2 to 3, etc...} . You could keep up the one-to-one correspondence forever.
Congratulations! You've just stumbled upon the smallest type of infinity (fancy term: "aleph-null").
IV. Why Infinity Plus One Isn't Bigger Than Infinity
So let's get back to why that commercial with the guy in the suit was wrong. To the casual observer, it would stand to reason that "aleph-null plus one" would be bigger than plain old aleph-null, but when we use the logic of set correspondence, we find that's not actually the case.
Let's take another example. I work at a radio station, so let's use the example of a big juicy microphone as our "plus one." We'll revisit SET A and SET B again, but this time, SET A has one extra thing in it:
SET A: {microphone, 1, 2, 3, 4, 5 ...}
SET B: {1, 2, 3, 4, 5, 6 ...}
Using correspondence, we'll match the microphone to 1, 1 to 2, 3 to 4, and so on. Visualized this way, you'll see it's possible to keep up this one-to-one correspondence between our sets forever, which means infinity and infinity plus one are actually equal. Georg Cantor, the mathematician who pioneered the work on infinity, said this logical contradiction essentially blew his mind. And as Alasdair Wilkins notes, "it gets weirder."
Imagine you built a set using only natural numbers ending in zero {0, 10, 20, 30, etc... }, and you compared that to a set using all the natural numbers {0, 1, 2, 3, etc...}. You'd think the second set would be ten times larger than the first set, but since both sets never end, as far as set theory is concerned, the two are equal. Crazy, right?
V. An Infinity Bigger Than Infinity
Well, if that's the case, you may find yourself asking how any infinity could ever be bigger than another infinity. Enter the world of real numbers. (A real number is any number representing a quantity along a continuous line. Forty-two, 2.335436643, the fraction 5/6, -5, and pi are all real numbers.)
Now imagine two points on a line. As an actuarial friend of mine put it to me last night, "Between any two finite points, there is an infinite and uncountable set of numbers between those two endpoints."
That's a logical idea. Expressed another way, you could say, "Infinity is the sides to a circle." And that makes logical sense as well. But since were dealing with set theory, let's put these two examples to the test using set theory.
So far, all of the sets we've dealt with have been "countable," which means that all the terms can be associated with a natural number {0, 1, 2, 3, 4, 5, etc ...} To get to a bigger infinity, we need to come up with something that is uncountably infinite (i.e. the sides to a circle).
Alasdair Williams spells this out brilliantly by telling us to imagine a binary number system, in which all the digits in every set are either zero or one. He then illustrates those sets as decimal expressions of real numbers.
SET A: {0, 0, 0, 0, 0, 0} .000000 ...
SET B: {1, 1, 1, 1, 1, 1} .111111 ...
SET C: {0, 1, 0, 1, 0, 1} .010101 ...
SET D: {1, 0, 1, 0, 1, 0} .101010 ...
SET E: {0, 0, 1, 1, 0, 0} .001100 ...
Williams tells us to imagine those sets extend forever. He then poses a question: can the numbers of any one set be rearranged in such a way that they create an entirely new set not contained within the original infinite set?
To address this, mathematician Georg Cantor proposed using what he called "diagonals." Namely, move through the sets diagonally, taking the inverse of each number and creating an entirely new set. It sounds complicated, so let's visualize it:
SET A: {0, 0, 0, 0, 0, 0} .000000 ...
SET B: {1, 1, 1, 1, 1, 1} .111111 ...
SET C: {0, 1, 0, 1, 0, 1} .010101 ...
SET D: {1, 0, 1, 0, 1, 0} .101010 ...
SET E: {1, 1, 0, 0, 1, 1} .110011 ...
We now have an entirely new set {0, 1, 0, 0, 1 ... } or .01001, which we know to be a real number.
Now let's take the inverse of that sequence diagonal {1, 0, 1, 1, 0 .... } or .10110. Is that number part of any of the sets we just created? It can't be a part of SET A because the 1 is opposite the 0, right? It can't be part of SET B because the 0 is opposite the 1. It can't be part of SET C because the 1 is opposite the zero. You see where this is going. The contradiction extends forever to infinity.
Conclusion: It's impossible to create a one-to-one correspondence between counting numbers and every possible real number. Thus, we've stumbled upon an infinity that is logically bigger than the infinity alpeh-null. Hooray! Using sets we've proven the idea that "between any two finite points there is an infinite and uncountable set of numbers between those two endpoints."
Take a few minutes to gather your brain and be sure to bring a paper and pencil to your next dinner party. When infinity comes up, start doodling sets and you're sure to be the resident expert who wows all the other guests with your l337 knowledge of epic maths. (I wouldn't recommend this as a sound way to pick up a date, but hey ... maybe it will work.) Good luck!
SOURCES/FURTHER READING:
As noted over and over, this article is heavily indebted to Alasdair Wilkins' wonderful post for Gizmodo, "A Brief Introduction to Infinity." Continue reading there if you want to wade deeper into infinity with his summary of "the continuum." I used a lot of Williams' set examples for this blog post (and as Colin and I prepared for last week's show).
I'd also recommend "The Hilbert Hotel" by Steve Strogatz, which outlines the work of mathematician Georg Cantor in a way much more eloquent than anything I would dare summarize.
This video breaking down the different types of numbers is fun.
If you're looking for an exploration of infinity in time, check out this explanation of "Planck Time."
Information about "Humphrey" was sourced, as always, from Muppet Wiki. And last, but certainly not least, check out The Colin McEnroe Show infinity edition with Steve Strogatz and astrophysicist Ron Mallett for more on sequences and infinity in the cosmos.
