Depends on your starting point. You can prove 1+1=2 quite easily from the Peano axioms: 1+1=1+S(0) [definition of 1] 1+S(0)=S(1+0) [definition of addition, a+

Depends on your starting point. You can prove 1+1=2 quite easily from the Peano axioms: 1+1=1+S(0) [definition of 1] 1+S(0)=S(1+0) [definition of addition, a+S(b)=S(a+b)] 1+1=S(1+0) [transitivity of equality] S(1+0)=S(1) [definition of addition, a+0=a] 1+1=S(1) [transitivity of equality] S(1)=2 [definition of 2] 1+1=2 [transitivity of equality]