#inductive set

Anyways so let's set about making some numbers! Along the way, we will uncover another axiom. It's all just fun and games here today!

Let's start with a little informal discussion first before we get into it. As it stands right now, the only thing that exists in the universe is sets. The most basic one is, of course, ∅ the empty set. We can then use the various axioms to create new sets; for example, we get {∅} from the pairing axiom, and {∅, {∅}} from the power set axiom...and so on. So we've built kind of a universe of sets that are just sets of sets of sets of sets ... of the empty set. How might we use these sets to model the natural numbers?

Here's the proposal:

0 = ∅

1 = {0} = {∅}

2 = {0, 1} = {∅,{∅}}

3 = {0, 1, 2} = {∅,{∅},{∅,{∅}}}

So in effect, every number is defined recursively as the set of all numbers preceding it. The cardinality of each set also coincides with our normal notion of what each number means. Curiously, we get from this construction that each number is both an element and a subset of the succeeding number. Interesting. This property actually turns out to be kind of useful later. So useful that it's going to get its own definition. But that's going to come in a later post.

For now, let's notice that the way we have constructed each natural number a+1 so far can be described as a+1 = a ∪ {a}. In fact, for any set a, its successor a+ is defined to be a ∪ {a}. We can then describe a set A being closed under successor, or inductive, as a set that for all members a ∈ A, we have a+ ∈ A.

Unfortunately, as of yet, we have no evidence that any such set exists. To be quite honest, we've only been looking at finite sets, and it seems that a set would need to be infinite to be inductive (besides the empty set, of course). Well, that's an easy enough fix: let's just axiom this problem away! The infinity axiom asserts that there exists an inductive set A with ∅ ∈ A, i.e., A is nonempty.

We can see that natural numbers we proposed at the beginning have been successors of each other, and throwing all these natural numbers into one set would give us an inductive set, almost by definition. In fact, we will actually proceed to define the natural numbers (finally!) in this way: a natural number is an element that belongs to every inductive set. From the infinity axiom and the subset axiom, we can be sure that the set containing the natural numbers, or the set of natural numbers is indeed a well-defined set that most definitely emerges from our axiomatic approach.