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

In 1889, Peano published the list of axioms that is now named after him. He thought that his recursive definitions of addition and multiplication were intuitively and obviously consistent, offering no justification at all for it:

He evidently thought that these equations in themselves define the addition and multiplication functions uniquely. He failed to realize the necessity for a proof that there are indeed functions satisfying them. There is therefore a lacuna in Peano's account at just this point.$^{[1]}$

When Landau says that “$x+y$ has not been defined”, he's criticizing Peano for not justifying formally that his recursive definition of $+$ is correct as far as existence and unicity are concerned. More generally, Landau is saying that such recursive definitions should have a rigorous foundation, which is the Recursion Theorem that was published by Dedekind in 1888. This is the theorem that Peano dismissed in his 1889 publication. In fact, Landau himself dismissed it in the first draft of his book:

It is a mark of the significance of Dedekind's achievement that Peano was not the only mathematician of stature to miss the need for it: Landau, for example, omitted the proof from the first draft of his account of arithmetic in 1930 and was saved from error in the published version only by the intervention of a colleague.$^{[1]}$

The statement of the Recursion Theorem is the following.

Recursion Theorem (Dedekind, 1888). Let $A$ be an arbitrary set containing an element $a\in A$, and a given mapping $g:A\to A$ of $A$ into itself. Then there is one and only one mapping $\phi:\mathbb{N}\to A$ with the two following properties: $$ \phi(0)=a\tag{1}, $$ $$ \phi(n+1)=g(\phi(n))\tag{2}. $$

A proof (essentially by induction) of this theorem can be found at pp. 16-17 of [2].

What misled Landau was no doubt the following short but fallacious argument for Dedekind's result. Equation $(1)$ enables us to define $\phi(0)$ uniquely. And if $\phi(n)$ is defined uniquely, equation $(2)$ enables us to define $\phi(n + 1)$ uniquely. Hence by induction the two equations together define $\phi(n)$ uniquely for every natural number $n$. The fallacy in this argument lies in the potential ambiguity of saying that $\phi(n)$ is ‘defined uniquely’ in advance of a proof that the function $\phi$ exists.$^{[1]}$

Since the Recursion Theorem is general, it can be used to justify the definitions of sum and product of natural numbers. However, it doesn't appear in Landau's book. Instead, more direct (and simple) proofs of existence and unicity for $+$ and $\times$ are given. These have been found by Dr Kalmár, as Landau confesses in his preface.

Sources:

[1] Potter, M. (2000), Reason’s Nearest Kin. Oxford: Oxford University Press.

The three excerpts are from pp. 82-83.

[2] H.-D. Ebbinghaus, H. Hermes, K. Lamotke, H. L. S. Orde et J. H. Ewing (1991) Numbers. Graduate texts in mathematics, Springer.

The original version is called Zahlen.

Answer 2

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 3

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 $x'$.