Second Ode on Induction: The Natural Numbers

Last time we went over an example of induction. Yes, even after I promised that we’d dispense with the examples and really dig in to the meaning behind induction. I did that because I wanted to show you something a little more complicated and a little more computer sciencey than the usual examples you see in discrete math for programmers, like proving that $\sum_{r=0}^{n}r = \frac{n(n+1)}{2}$. Binary trees are fairly simple and very computer sciencey, so I thought they’d be nice. I know I didn’t see a decent induction proof involving a data structure until my upper-division algorithms class, and that’s just too late, in my opinion.

Anyway, I said last time that the three parts of induction could be explained in terms of an argument you make to your friend when he says he was abducted by aliens who stripped him naked and looked at him. Like this:

1. Base case: “Here’s an alien from the species you say abducted you, let’s call them the Borg. This Borg says he personally would never travel across the entire galaxy from the Delta Quadrant just to see you naked.”
2. Inductive hypothesis: “Let’s assume that if you polled $n$ Borg drones, they would agree with this drone—they wouldn’t fly across the galaxy to see you naked.”
3. Inductive conclusion: “Given that $n$ drones concur, $n+1$ drones concur. Therefore the entire collective concurs. The Borg would never fly from the Delta Quadrant so they could abduct you and look at you naked.”

Why does the inductive conclusion work? Why can we assume that $n$ concurring drones means $n+1$ concurring drones, and that this property means a unanimous collective? If you’ve seen Star Trek, you know that it’s because the Borg have a hive mind. If one drone holds an opinion, that opinion is shared by the entire collective. But why does this work for induction?

The Natural Numbers

The natural numbers are just the normal counting numbers that you learned in kindergarten (1, 2, 3…). Zero technically isn’t a natural number, but sometimes it’s convenient to include it (our binary tree example from last time was one instance where it was useful).

Imagine that we lined up all the Borg drones in the collective in a big line that stretches off to infinity, and gave each of them a number, starting at 1.

The Borg can assimilate other species and make them into Borg drones, so the collective is always growing. Suppose a poor dumb Bolian gets assimilated and has to join the line, but we don’t want to make him walk all the way to the end, so we let him cut in right here in the middle, at position 498. Since we want our Borg line to be in order by number, we just give our new Bolian drone the number 498, and the other drones’ numbers all go up by one. No matter how many drones were in our line before we let the Bolian cut in at position 498, there are always enough natural numbers that we can bump everyone up by one without running out. And no matter how many new drones we allow to cut into the line at the middle, we can always repeat this move and bump everyone’s number up by one. Everyone always gets an index.

This is what mathematicians mean when they say the natural numbers are infinite: no matter how many numbers we assign to drones, there are always more of them. The natural numbers never run out.

The Peano axioms

But how do we know that bumping everyone’s number up by one doesn’t somehow change someone’s number into something that’s not a natural number? How do we know that we won’t bump up number 5354 and somehow end up with number 5354.12?

Your first reaction to that was probably “Of course we won’t. How could that happen? That’s stupid!” But mathematicians like to be extra careful about things like that. They like to prove that it can’t happen. But there’s really no obvious way to actually prove that in the mathematical sense. All we know is that if we have five mangos and we get another mango, we have six mangos; we don’t have 5.5 unless we cut the sixth mango in half and eat one half.

When mathematicians can’t prove something, but it seems obvious or self-evident that it should be true, they often make it into an axiom. Axioms are rules that we can’t prove, but that seem like they should be true, so we just assume they are and use them to prove things. Even the most skeptical mind can’t constantly be in doubt about everything, or knowledge would never get anywhere. For example, when scientists run experiments, they usually assume that they aren’t hallucinating the entire experiment from the inside of a padded cell. You can’t really prove this; we could all be in the Matrix right now. But it seems unproductive to worry about that, so we just assume that isn’t the case and go about our lives. In the same way, mathematicians are allowed to assume certain things are true, if it seems like it just couldn’t work any other way. Non-Euclidean geometry is an interesting example of what can happen if you change your axioms a little, but we won’t go into that here (maybe in a future post).

So there are certain things we just assume to be true about the natural numbers, certain axioms that define them. These are called the Peano axioms, after Italian mathematician Giuseppe Peano, who worked on mathematical induction. See Wikipedia for more information.

For induction, we mostly care about just three of the Peano axioms. Those are:

1. 1 is a natural number.
2. If $n$ is a natural number, then $n+1$ is a natural number.
3. If $K$ is a set which contains 1 and which contains $n+1$ whenever it contains $n$, then $K$ contains every natural number.

The first two give us a recursive definition (!) of the natural numbers. 1 is a natural number. Since 1 is a natural number, 1+1 = 2 is also a natural number. Since 2 is a natural number, 2+1 = 3 is also a natural number. So on. We can go as high as we want, but hopefully you see what’s happening; if we start with one Borg drone, labeled 1, and keep adding drones to the front of the line, the original drone’s label will keep going higher and higher. According to the second axiom above, no matter how high the first drone’s number gets, it’s always a natural number, because it was a natural number when we started and we just bumped it up by one.

Mathematicians often say that $n+1$ is the successor of $n$. Rather than saying that you add 1 to a natural number to get the next natural number, they think of it as applying a successor function, $S(n)$. This way of thinking about it is useful later, when we start applying the ideas behind the Peano axioms to things other than numbers, like binary trees.

The third axiom above is called the inductive axiom. It’s what allows us to write proofs by induction. But we need one more idea to see exactly why induction is valid: the isomorphism. We’ll go over that next time. Isomorphism will show us how we can create a set of natural numbers that represent some other object, like sums or binary trees, prove things about that set, and translate those proofs into proofs about our original objects.

Summary

• The natural numbers are infinite; no matter how many of them you have, there’s always another one.
• The natural numbers can be defined recursively by starting at 1 and taking the successor.
• One of the rules of the natural numbers, the inductive axiom, combines with the idea of isomorphism to give us proofs by induction.