(where x' denotes the successor of x)To every pair of numbers x, y, we may assign in exactly one way a natural number, called x + y (+ to be read "plus"), such that

1) x + 1 = x' for every x,

2) x + y' = (x + y)' for every x and every y.

x + y is called the sum of x and y, or the number obtained by the addition of y to x.

The following proof is quite hard to grasp (at least for me), so I'd be very grateful if anyone of you could post a link with a proof of this theorem or propose their own proof. Thanks very much.

P.S. If needed, I can post Landau's axioms & his proof.