1. The problem statement, all variables and given/known data Hi everybody, i'm going through the book Fondations of Analysis by E. Landau. I'm trying to prove theorems by myself and then checking if they are correct. But i proved this theorem in a different way then the book and i need a check. thank you Theorem 4 To every pair of numbers x,y we may assign an unique number x+y such that 1) for every x x+1=x' 2) for every x and y x+y'=(x+y)' 2. Relevant equations Axiom 2 for each x there exist an unique number called the successor of x, denoted by x' Axiom 4 if x'=y' then x=y 3. The attempt at a solution I proved existence in the same way of the book and so i know it's right. For uniqueness i did this. Let's take x,y to be arbitrary natural numbers. We assume there there exist z and w such that x+y = z and x+y = w and that they satisfies properties 1) and 2). By axiom 2 (x+y)' = z and (x+y)' = w' Then by property 2) x+y' = (x+y)' = z' = w' and by axiom 4 z = w The book prove this by induction, so maybe i missed something.