Today, I'm going to prove that 1+1=2. Why, you ask? Because I wasn't around last time the subject came up, and then it came up again, and now I am around.
We start with Peano's axioms. These are not arbitrary rules I'm pulling out of thin air, we need some axioms to make sure 1, +, =, and 2 mean something other than just being a set of lines on paper (or pixels on a screen), and in particular that what it means is the same we mean when we say "one", "plus", "equals" and "two". Behold:
- 0 is a natural number.
- For every natural number x, x = x.
- For all natural numbers x and y, if x = y, then y = x.
- For all natural numbers x, y and z, if x = y and y = z, then x = z.
- For all a and b, if a is a natural number and a = b, then b is also a natural number.
- For every natural number n, S(n) is a natural number. (S is called the "successor" function. It will be important.)
- For every natural number n, S(n) = 0 is False
- For all natural numbers m and n, if S(m) = S(n), then m = n
We will be ignoring the axioms relating to induction, because I am lazy.
Now, you'll notice so far we only have explicitly named one number, 0, but axiom 6 allows us to build more. We know that if n is a natural number, then so is S(n). Then, from 1 and 6 we know we have a natural number called S(0). And since S(0) is a natural number, so is S(S(0)). We could go on for as long as we wanted, but these will suffice.
But wait, how do I know I really do have different natural numbers, and not just 0 called by different names? Axiom 6 only specified that S(n) is natural, not that it's different from n. Fortunately, we have axioms 7 and 8 for that. 7 says S(n) cannot be equal to 0, so in particular S(0) and S(S(0)) are different from 0. How do we know S(0) is different from S(S(0))? With the help of axiom 8, which tells us that if S(S(0)) is equal to S(0), then S(0) has to be equal to 0. We already know that is not the case, so we have proven that 0, S(0), and S(S(0)) are all different numbers. We could give them different names, if we wanted to. The usual names for them are 1 and 2.
Now, we already have "1" [S(0)] and "2" [S(S(0))]. "=" has already appeared before and axioms 2 through 5 deal with how it works. So the only missing element is "+", the addition function. The addition function as we're gonna use it takes two natural numbers as arguments and outputs another natural number (in less words, that would be + : N × N → N), and we can define it (recursively) as:
- a + 0 = a
- a + S(b) = S(a+b)
Now that we know what every part of 1+1=2 means, we can say that:
1+1 would be written as S(0) + S(0)
By the second part of the definition of addition, S(0) + S(0) = S( S(0) + 0 )
By the first part of the definition of addition, S(0) + 0 = S(0)
Therefore: S(0) + S(0) = S( S(0) )
1+1=2
There you have it. To the extent that 1, +, =, and 2 are defined to be the same things we mean when we use the words, it can be proven that 1+1=2.
Now, I am not a mathematician. I know there are plenty of discussions about definitions and axioms and first- and second-order logic and ZFC set theory and who knows what else that go way above my head. But 1+1=2, and that can be proven.