Complete proof of 1+1=2

Question

I'm searching for the proof that

• Started from Peano's axioms
• Using modern symbols
• Detailed as possible

Could you give me a link to it? Thanks

Answer 1

There isn't a lot to prove here. Either you define $2$ to be $1+1$, so $2 := 1+1$, or you define it to be the successor of $1$, so $2:=S(1)=S(S(0))$, where $S\colon\mathbb N\to\mathbb N$ is the successor function. But the definition of $+\colon \mathbb N\times\mathbb N\to\mathbb N$ is a recursive one with $a+0 := a$ and $a+S(b):=S(a+b)$, so that $$ 1+1 = 1 + S(0) = S(1+0) = S(1) = 2. $$

Answer 2

Well, starting from the Peano axioms define the addition as

$m+0 =m$ and $m+n'=(m+n)'$,

where $n'$ denotes the successor of $n$ according to the axioms.

Define $1=0'$, $2=1'$, and so on.

Then it follows that $1+1 = 1+0' = (1+0)' = 1' =2$. Done.