Can you prove that 1+1=2?

If you define the natural numbers with the Peano-axioms, you can define the addition of two numbers (recursively) as:

1. n + 0 = n

2. n + m' = (n + m)'

Where n' is the successor of n.
So in this case we have 1 + 1. With 1 being the successor of 0, so 1=0' we can write that as:

1 + 0'

With the second equation, we get:

1 + 0' = (1 + 0)'

With the first equation now, we get that

1 + 0 = 1

, so:

(1 + 0)' = (1)' = 1'

And we know the successor of 1, which is 2. Thus:
1 + 1 = 1' = 2