Landau's "Foundations of Analysis" - Addition of natural numbers

Question

At the beginning of his Foundations of Analysis book (translated from German), Landau writes in his Preface for the Teacher :

Peano defines $x+y$ for fixed $x$ and all $y$ as follows : $$x+1 = x' \\ x+y' = (x+y)',$$ and he and his successors then think that $x+y$ is defined generally ; for, the set of $y$'s for which it is defined contains $1$, and contains $y'$ if it contains $y$.

But $x+y$ has not been defined.

All would be well if - and this is not done in Peano's method because order is introduced only after addition - one had the concept "numbers $\leq y$" and could speak of the set of $y$'s for which there is an $f(z)$, defined for $z \leq y$, with the properties $$f(1) = x, \\ f(z') = (f(z))' \quad \text{for } z < y.$$

Here the prime denotes the successor function. I really don't understand why $x+y$ can't be defined the first way. In fact, Theorem 4 of the book, which is at the same time Definition 1, seems to follow exactly Peano's definition. But, the author actually proves unicity and existence of such a function $+ : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$. He fixes an $x$ and shows unicity and existence for all $y$, so that given $(x,y)$, $+(x,y)$ is well defined.

The only difference I see between Peano's and Landau's definitions of natural numbers is that Landau's proves existence by induction on $x$ instead of $y$.

What's the subtlety here ? Why is Peano's definition incorrect ?

Answer 1

Something such as the iteration theorem should be proved first:
(P,S,1) is a Peano system
W is a set, $c \in W$, $g : W\rightarrow W.$
Conclusion: There exists a unique function $F: P \rightarrow W$ such that
(a) F(1)=c
(b) F(S(x))=g(F(x)).

One can then apply the iteration theorem to $W=P$ and $g=S$ in order to obtain the existence of a unique binary operation + on $P$ such that
(a) x + 1 = S(x)
(b) x+S(y)=S(x+y) for all $x,y \in P$.

Answer 2

I guess the problem is that Peano only defines (by induction over $y$) a function $x+ : \mathbb{N} \to \mathbb{N}$ for an arbitrary (but fixed) $x\in \mathbb{N}$, and misses an argument for how to combine these infinitely many definitions into one function $+: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$.

Landau instead defines (by induction over $x$) the functions $x+ : \mathbb{N} \to \mathbb{N}$ for all $x \in \mathbb{N}$, (he uses induction over $y$ to prove uniqueness of each such function), and because he defined all these functions by induction over $x$, he is able to combine them into one function $+: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$: the set of $x$'s for which this function is "defined for all $y$" contains $1$, and with $x$ contains $x'$.