(0, 1) = {1, 2}

In higher math, everything is defined in terms of sets. I’ll show how the first few primitives are defined. I think this might go over well with my mostly-programmer audience 🙂

The integer zero is defined as the null set, {}. Then, each subsequent integer is the set of all previous integers. This has the cool side effect of meaning each integer has cardinality equal to its “value,” while preserving the requirement that a set can’t contain the same element more than once.

0 = {}
1 = {{}} = {0}
2 = {{}, {{}}} = {0, 1}
3 = { {}, {{}}, {{}, {{}}} } = {0, 1, 2}

One thing that’s interesting about sets is that order doesn’t matter. {1, 2} = {2, 1}. So what do you do if you want an ordered pair? Well, those are defined as follows:

(a, b) = {{a}, {a, b}}

So, you can see that
(a, b) != (b, a)
{{a}, {a, b}} != {{b}, {a, b}}

Now, check this out:

(0, 1)
={{0}, {0, 1}}
={1, {0, 1}}
={1, 2}

ta-da! I think you have to be like 3 levels of meta-nerd before you think that’s cool. Also, I think that’s cool.

In case you’re interested, each rational number is defined as all pairs of integers whose quotient is the rational. For example,
1/2 = {(1, 2), (2, 4), (3, 6), ...}
9/8 = {(9,8), (18, 16), (27, 24) ...}

And the real numbers are defined as the set of all rational numbers less than the real number. I won’t try to illustrate that here!

Also, I don’t remember how negative integers are defined, but if you have that you can define negative rationals and from those define negative reals.

Programmers, you may realize at this point that this means the integer 0 is not the same as the rational number 0! A strange parallel to programming types, no?

As an exercise to the reader, why not try defining complex numbers? A good first step might be defining the Gaussian Integers. Post your answers in the comments! If there are no comments, I will know that everyone who reads this is a wuss (which might be vacuously true 😦 ).


3 Responses to (0, 1) = {1, 2}

  1. Ben says:

    Complex numbers are often thought of as ordered pairs, like they are in the complex plain, but with an extra bit of information saying “hey guys, this is an imaginary number!”. With that basis for my reasoning, you just need two things: Ordered Triplets in set notation and i in set theoretic notation.

    The first (ordered triplets) follows recursively.
    (a,b,c) = { a, (b, c) } = { a, {b, {b, c}} }

    Defining i in set terms was trickier and in fact I have punted it because I want to go write something myself.

    You made me write set notation with pencil-and-paper for the first time in a few years. Happy.

    • joblivious says:

      Close re triples! If you say (a, b, c) is {a, (b, c)} then (a, b, c) = {a, (b, c)} = {(b, c), a} = ((b, c), a). What you really need is (a, b, c) = (a, (b, c)) = {{a}, {a, {{b}, {b, c}}}}

      As for complex numbers, I don’t know the answer either. I thought I did but your post made me realize I don’t!

  2. Ben says:

    I think the easiest thing to do with i would be to define the operations up to square roots and then just take sqrt(-1).

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