Proof of Edmund Landau's Foundation of Analysis

Question

The theorem is as following:

The proof was split into two parts, namely: Uniqueness and Existence, I have hard time understanding the existence part:

Namely: in the 10-12th line, what did Edmund Landau mean by saying : "$Then\; the\; number\; x'+y=(x+y)' \;is\; the\; required\; number\; for\; x'$" Isn't this similar to what we are trying to prove? namely: $x+y'=(x+y)'$ Did he assumed it? If so, why are we allowed to that? (why don't we just assumed Theorem 4 right away, if this was the case) Thanks!

Answer 1

He uses an inductive proof here. To be specific, on the outside, he does induction on $x$, and shows that $x$ has the desired property for any $y$.

Part I of his proof is of course the inductive base $x=1$ for this proof. That is, he shows that for $x=1$ there exists a number with the desired property given any $y$.

Part II is the inductive step. Here, he shows that if for some $x$ there exists some number $x+y$ that has the desired property given any $y$, then for $x'$, there exists some number $x'+y$ that has the desired property given any $y$.

In particular: the inductive hypothesis is that for some $x$ it is true that:

1) $x+1=x'$

2) $x+y'=(x+y)'$

And he uses that to prove:

1) $x'+1=x'$

2) $x'+y'=(x'+y)'$

So, when you say:

Isn't this similar to what we are trying to prove? namely: $x+y'=(x+y)'$

He assumes $x+y'=x+y$, but he is trying to show that $x'+y'=(x'+y)'$. Those are similar, yes, but they are (crucially!) not the same.

Did he assumed it? If so, why are we allowed to that?

Yes: he assumed $x+y'=(x+y)'$, but yes: he is allowed to do so, since that is the nature of inductive proofs.

Hope that helps!