evolution

It’s time to begin down the path to real numbers, but we have some things to do first. Namely, we’ll construct the natural numbers using basic concepts related to sets. We shall assume the validity of the standard (Zermelo-Frankel with choice) set theory, which is known believed to be a consistent logical system as discussed previously. To enhance readability(!), I shall paper over some subtleties. I’ll assume that you know (at least intuitively) what a set is, and that you know what the union of two sets is (the collection of all elements in both sets; it helps to recall the Venn diagrams you did in school).

We are all apparently innately familiar with the natural numbers. So, we have some idea where we are going. Any model set we hope to build that does what we intend must accomplish the following:

1. Every member of the set must have a “successor”, i.e., given an element of the set, there must be a way to construct the elements that “come after”. These elements are also members of the set.
2. There is one element that is not the successor of any other element. We shall christen this element 0. (This is just a name, it need not correspond to what we know as zero. But, the only other number we know of that would work is 1.)
3. Elements with the same successor are equal.
4. Any subset of this model set which contains our 0, and given any element the subset also contains its successor, then this subset must in fact be the whole set. You may have heard this called the “induction principle”.

It has been proven that at least one set with these properties does in fact exist, assuming the validity of set theory.

Very well, let us plod on. Throughout, we denote the empty set ∅ and the phrase “defined to be equal to” by “:=”. Then, define:

• 0 := ∅
• 1 := {∅} — “a box containing an empty box”
• 2 := {∅, {∅}} = {0, 1}
• 3 := (∅, {∅}, {∅, {∅}}} = {0,1,2}

Let us define the successor S(a) of a element a to be a ∪ {a}. I claim that this construction satisfies the conditions outlined above (known as the Peano axioms).

All right — so how do you “add” two of these things together? Define an operation called “addition” and denoted by + as follows:

• a + 0 = a; and
• a + S(b) = S(a) + b

This is an example of a recursive definition; i.e. a definition that is applied repeatedly until one arrives at the first of these statements, and then unwound from there. For example, substitute 3 for a and 4 for b. One gets from the definition that 3 + 4 = 3 + S(3) = S(2) + S(3) = S(2) + S(S(2)) = S(S(1)) + S(S(S(1)), and so forth.

This operation we call “addition” is commutative; i.e., a + b = b + a for any choice of a or b. 0 is also what we call an “identity” element; by definition a + 0 = a for any a. Furthermore, 0 together with addition “generate” the entire set — remember, if the set contains an element, it also contains the element’s successor. (In jargon, this set with addition is a “free monoid on one generator”.)

You can define “multiplication” thusly: a * 0 = 0 and a * S(b) = (a * b) + a. This is also a recursive definition. You can test it if you like by substituting 3 for a and 2 for b. If you apply the definition correctly, you will get the expected result. (The set with multiplication is a “commutative monoid with identity 1.” The set with both operations together is a “commutative semi-ring”.)

With some effort it can be shown that these operations satisfy the expected properties of arithmetic, like the distributive law.

The mathematician Leopold Kronecker asserted that “the natural numbers are the work of God, all else is the work of man.” Here, we see that it is not required to assume their existence, although even infants have been shown to grasp the concept of “two” or “three”. They may be constructed from scratch out of a consistent logical system that does not assume their existence.

The question of whether this construction and the theory behind it expresses some fundamental concept inherent in our existence or is simply a word-and-symbol game is another matter entirely.

[References: Wikipedia, Kenneth Ross's accessible Elementary Analysis: The Theory of Calculus, which basically continues where I left off here. It had been a while since I thought about the ideas in this post! UPDATE: And thanks once again to David's fact-checking.]