In this post I’ll show how Set Theory allows us to define the natural numbers using sets. It’s admittedly a very abstract topic, but it’s about something very common in our experience: counting things. Seeing how numbers are defined also demonstrates (contrary to some false notions) that there is a huge difference between a number and how that number is “spelled” or represented.
Note: I am not a mathematician! This topic is right on the edge of my mathematical frontier. I wanted this addendum to the previous post, but be aware I may misstep. I welcome any feedback from Real Mathematicians!
But go on anyway… keep reading… I dare ya!
We’re used to using numbers to count things. For example, if we’re playing penny-poker, we know how many pennies we have sitting in front of us. We’re also used to comparing things we’ve counted; we know that 27 pennies is more than 22 pennies.
The question is: how do we know these things? What allows us to count, one, two, three? Is that the same as 1, 2, 3? What about I, II, III? What tells us that seven (or 7) is greater than three (or 3)? It’s the basic math we learned in school, but where does it come from? What makes it work the way it does? How do we know it works right?
It all starts with Set Theory.
[Full disclosure: not all mathematicians use set theory to define the natural numbers; there are other ways to formally define them. Even so, these alternates share the common approach described below of an initial number and a successor function.]
When we define any mathematical system we start with a small number of rules, or axioms, that we take as the core truths of the system. These are the “givens” of the system. (As in, “given X, therefore Y.”) Everything else in the system (theorems, the “therefores”) must be built from the axioms.
The geometry you learned in high school — Euclidean geometry with the right angles and circles and lines — has at its core a small set of axioms upon which everything else is constructed. (I was always fascinated by how much you could do in geometry with just a straight-edge and a compass.)
In the previous post I wrote about how sets have members. And they have a size, which is a count of how many members are in the set. Regardless of what kind of members a set has, we can always compare sets based on their size. A set of 10 pennies and a set of 10 elephants are identical (in size).
In fact, when talking about set sizes, we don’t care what the members are. We only care about how many (the set size). Pennies, elephants, family members or whatever, it doesn’t matter.
To create a formal definition of (natural) numbers, we start with a set that has no members. We call this the empty set. You can think of it as a grocery bag with nothing in it. This empty set is the first axiom. It’s a given, and we equate it with the number “zero.”
Venn diagrams are just set theory in action!
Mathematicians use curly braces, {…}, to notate sets. What’s between the braces is what’s in the set. For example, the set of the first three English capital letters is: {A, B, C}. (Commas separate members.)
The empty set is notated as simply: {}. (I’m going to use different colors for the sets to help keep things clear.)
The second axiom is that we define a successor function, which is a way, given some thing, to generate the next thing in line. (In a sense, ‘having children’ is a successor function to generate the next generation in your personal family tree.)
For the natural numbers, we define a successor function to — given some set (number) — generate the next set (number) in line. We do that by saying that the next set in line is a set containing all the previous sets.
So far, we only have one set, the axiomatic empty set, which we’re calling “zero.” To create the next set (number), we create a set containing all previous sets, which in this case is only the empty set. That gives us a set that looks like this: {{}}.
Think of it as putting an empty grocery bag inside another grocery bag. Now you have a grocery bag containing a grocery bag. We’ll call this new set “one.”
Now, because we defined the first set, {}, as 0 (“zero”), we could also write the new set (“one”) as: {0}. It might seem odd that “one” sort of looks like a “zero” with curly braces around it, but our successor function says that any given set (number) contains all previous sets (numbers), so the “one” set contains everything except the “one” set.
Note an important thing: the set for “one” contains one member!
Let’s keep going. The next set, which we’re going to call “two” needs to contain the previous sets. We’ve defined “zero” and “one”, so one way to show “two” is the set: {0, 1}. (Note that it has two members!) The formal way to show it looks a bit confusing: {{}, {{}}}. Just remember that {}=0, and {{}}=1.
So now that we have {{}, {{}}}=2, we can define “three” as: {{}, {{}}, {{},{{}}}}. We can also write this as: {0, 1, 2}. And, again, note this set has three members.
We can continue using the successor function to create all the natural numbers. The formal way gets hard to follow as the curly braces proliferate, so from now on, we’ll use the way that uses the digits. For example, the number “nine” is notated as: {0, 1, 2, 3, 4, 5, 6, 7, 8}. (Which has nine members.)
[You might wonder if it wouldn't be easier to just fill each new grocery bag with X number of empty bags. For instance, "three" would be: {{}, {}, {}}. It does create sets with increasing sizes, but breaks the successor rule of "each next set includes all previous sets." There are some important set-theoretical reasons for that rule.]
A very important point here is that, although we’ve said {}=0, {{}}=1, and so forth, the actual number symbols (1,2,3,…) or their names (“one”, “two”, “three”,…) are not part of the definition! This is crucial to understand! The definition of the natural numbers is strictly based on set sizes.
Think of this in terms of grocery bags or piles of pennies. All we care about is that a given set has a given size. These set sizes have a variety of ways they can be expressed, and all such expressions are equal. For example:
{{}} = one = 1 = I = uno = ein = ένας = एक = satu = один
These are all various ways of “spelling” the number of noses most people have on their face. (My apologies to those with a lesser or greater number.)
There is a world of difference between a numeric quantity and the way that quantity is notated or “spelled.” This is rather obvious when it comes to different languages (and hopefully Google Translate did me right on the last four above).
This may be less obvious when it comes to the familiar (base ten) notation of digits that most of the world uses for basic math. I’ve actually heard of people who think that our base ten system is somehow “special” or different from other bases.
It’s not. It’s purely coincidence based on the usual number of fingers most people have. Other cultures have, in fact, used other number bases. And when it comes to counting things, no number base is any better than any other (save, maybe, for how many symbols you have to memorize and for the length of the number string).
But I think that’s enough for today. Next time I’ll continue with number bases, and if you stick with me, you’ll walk away understanding “binary” and “octal” and “hexadecimal” number systems.
Summary: Every natural number is defined as a set that contains all previous (“less than”) sets (numbers). The size of the set is the number! In a very real sense, each number is just a size (formal term: cardinality) of the set representing it. Numbers have a variety of — equivalent — ways of notating or “spelling” them.
