We have seen in another post that set theory is a good candidate to most fundamental theory of the universe. One of the reasons for it is that the rest of mathematics can be derived from it.
Let’s see, as an example and mainly for fun, how to multiply two natural numbers using set theory, that is, how to arrive to arithmetic from set theory.
We can make some basic operations with sets. We have union, intersection, symmetric difference, cartesian product and powerset among others. In order to infer multiplication we need to define cartesian product.
Given two sets, A and B, their cartesian product is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. An example:
A = {1, 2, 3}
B = {red, blue}
A x B = {(1, red), (1, blue), (2, red), (2, blue), (3, red), (3, blue)}
We also need another concept: cardinality. It’s just the number of members that a set has. The cardinality of A is 3, the cardinality of B is 2… you get the gist. It’s represented by two vertical bars, like this: |A|.
Now we need to define natural numbers as sets. In order to achieve this, we’re going to make up some new sets, that we’re going to call Z, O, T and H, which are short names for Zero, One, Two and tHree.
Z = {} = ∅ = 0
O = {0} = {∅} = 1
T = {0, 1} = {∅, {∅}} = 2
H = {0, 1, 2} = {∅, {∅}, {∅, {∅}}} = 3
This is the same as saying that each number is the set of all the preceding numbers. It’s okay if you want to ponder it a little, I’m still pondering it.
Remember cardinality? This is the cardinality of our new sets:
|Z| = 0
|O| = 1
|T| = 2
|H| = 3
Let’s make the cartesian product of sets T and H. We’re going to call the result of this product (you guessed it!) S:
S = T x H = {(0, 0), (1, 0), (0, 1), (1, 1), (0, 2), (1, 2)}
Let’s find out the cardinality of S:
|S| = 6
Yay! We have multiplication!
If you have read this further, you are not allowed to ask what use can this possibly have.