This blog post uses peano axioms to proof that 1+1=2. In case you were not sure that one and one equals two.
We use the following definitions:
- $0$ is a natural number
- $1 = S(0)$
- $2 = S(1)$
With $S$ the successor function. For the number $0$ and the successor function, the existence of them is enough, their exact definition does not matter. $1$ and $2$ on the other hand, are fully stated by the definitions of $0$ and $S$
Now, let's define addition.
It is an operation with the following two properties:
$$a + 0 = 0\\a + S(b) = S(a+b)$$
So this means:
$$1+1=1+S(0)=S(1+0)=S(1)=2$$
QED.
We can use this pattern to calculate fancier things, like $2+2=4$
$$2+2=2+S(1)=S(2+1)=S(2+S(0))=S(S(2+0))=S(S(2))=S(3)=4$$
And all other kind of additions.
Further, we can define multiplication as
$$a\times 0 = 0\\a\times S(b) = a + (a\times b)$$
Now is
$$1\times 1 = 1\times S(0) = 1+ (1\times 0)=1+0=1$$
and
$$2\times 2 = 2\times S(1) = 2+ (2\times 1)=2+ (2\times 1) = 2+(2 \times S(0)) = 2+(2+(2 \times 0)) = 2+(2+0)=2+2=2+S(1)=S(2+1)=S(2+S(0))=S(S(2+0))=S(S(2))=S(3)=4$$
QED