Not Your Ordinary One Plus One Proof

“Sige nga, one plus one?”

“Two!”   

“Paano naging two?”

“One plus one” questions are quite a classic in one’s school life. Some ridiculous answers about such questions include “eleven,” “window,” and even a sarcastic remark such as “that can’t be answered because there’s no equal sign.” When asked to prove that one plus one is equal to whatever they answered, there are also quite a number of approaches. Some will actually draw a window; some will just put two one’s together, while some will just say “that’s just common sense.”

Well, if we take the question seriously, well, the answer to this immortal question is “two;” but why two? Some people put their index fingers together to prove that one and one is two, while others just scoff at such a worthless question. Ridiculous or worthless this thing might be, this irritating yet somehow interesting question gave rise to what could be considered as one of the most disturbing mathematical proofs that left people shaking their heads, wondering why one spent a lot of time proving something that can be done with one’s fingers alone.

What is this mathematical proof, anyway? Well, this proof doesn’t have a definite name, but it does employ one of the most obscure mathematical postulates ever created, governing a very basic branch of mathematics; the number theory. This set of postulates is better known as the Peano Postulates.

Created by mathematician Giuseppe Peano, this set of postulates describes how natural numbers really work. Natural numbers are numbers that exist in the real world; hence the name. Natural numbers include 1, 2, 3, 4, and so on. Arguments are still ongoing to determine whether zero is a natural number or not (although the Peano Postulates seem to consider zero as a natural number), but don’t fret; it will not disturb our discussion whatsoever.

Anyway, since the Peano Postulates govern the behavior of natural numbers, then we can use it to prove a mathematical sentence that involves adding two natural numbers that result to another natural number. We begin the proof by stating N as the smallest possible set of natural numbers that satisfy the five postulates below, which are derived from the principles of the Peano Postulates. So as not to complicate things, this proof assumed that zero is NOT a natural number.

Postulate 1:  1 is in N.
This couldn’t get any simpler; P1 says that 1 should always be in N.

Postulate 2: If x is in N, then its "successor" x' is in N.
For example, x is two. We know that the number that comes after two is three. So x’, x’s successor, is three. Since three is also a natural number, x’ is also in N. This is basically what P2 is all about.

Postulate 3:  There is no x such that x' = 1.
x’ is x’s successor. So if x’ equals one, x should be zero. But then again, as what was aforementioned, for the sake of convenience, this proof disregarded zero as a natural number.

Postulate 4:  If x isn't 1, then there is a y in N such that y' = x.
This is an extension of Postulate 3. If y is to exist in N (the way I call the number doesn’t matter; I could’ve called it “hoopla” instead of “y”), x cannot be 1. For example, x is five. According to P4, there exists a y such that it’s successor, y’, is equal to x, which happens to be five. With this said, we can say that y is four, which is definitely a natural number, and thus really belongs to N. However, if we force that x be equal to 1, then, according to P4, y is zero, which is in conflict with P3. x therefore, cannot be 1.

Postulate 5:  If S is a subset of N, 1 is in S and the implication (x in S --> x' in S) holds, then S = N.
This postulate isn’t really necessary in proving that one plus one equals two, but this is worth mentioning nevertheless. P5 says that if there is a set S which is a subset of N (N retains all of its aforementioned properties), 1 is also in S (reminiscent of P1) and S observes the second postulate (that a natural number and its successor are both in a set of natural numbers), then S is N, which makes sense. They obey the same set of postulates, and that makes them exactly similar.

Now why do we need to mention all these stuff? This is because our every move from this point must be based on those five postulates to be consistent with our proof. And now, we shall demonstrate addition, applying the five postulates we’ve just discussed.

Let’s give two variables that represent natural numbers; “u” and “i” (pun intended), “a” and “b,” or “x” and “y.” I prefer the last pair, so let’s go with them.

Next, I shall set y equal to 1. It doesn’t matter which variable is equal to 1. I just want y to be equal to 1, that’s all. So, if y is equal to one, then x + y --> x + 1 --> x’! If you add 1 to a natural number, you get the successor of that natural number. Now, we can say that x + y = x’, if y = 1.

Now what? The path to proving “1 + 1 = 2” is laid bare! We know that adding 1 to a number gives us that number’s successor. Thus:

If x + y = x’ if y = 1, then 1 + 1 = 1’ or 2 (QED)

Now you might wonder whether this proof only works on “one plus one.” What if y isn’t equal to 1? Then we apply the fourth postulate. A third variable is going in; let’s call the dude “z.” The fourth postulate implies that if y isn’t equal to 1, a number z exists, whose successor is equal to y. We now need to construct the mathematical sentence; but how?

Let’s call the sum of x and y as... “sum.”
x + y = sum

We know that z’ = y. So:
x + z’ = sum

We also know that z’ is z + 1. Hence:
x + z + 1 = sum

Finally, we know that when we add 1 to any number, we get that number’s successor. Thus:
(x + z)’ = sum

This follows that: x + y = (x + z)’ (QED)

The only way to test our proof is to plug in a couple of numbers. Let’s try 2 and 6. We know that 2 plus 6 gives us 8. Let’s see whether our proof is consistent with this logic.

x + y = (x + z)’
2 + 6 = (2 + 5)’
2 + 6 = 7’
2 + 6 = 8

I think this proof is enough. Such is the disturbing nature of number theory, its curious ability to turn docile mathematical sentences into hellholes. Anyway, I hope I’ve enlightened you about the fact that there’s more to proving simple addition than putting your fingers together. And now I ask you; what is one plus one?

Note: The original proof can be found on this site: http://mathforum.org/library/drmath/view/51551.html
I've decided to explain the proof further through this article to dilute its difficulty.