How is addition on N formally defined in textbooks on real analysis?

Question

This is a follow-up question to Why does the definition of addition require proofs? In Landau's Foundations of Analysis, his definition of addition on the natural numbers seems a bit strange to me -- this '$+$' operator seems to pop out of thin air. I'm wondering if this "definition" is typical of textbooks on real analysis these days. How else is it being defined?

Answer 1

On the contrary, it doesn't "pop out of thin air" at all; I think you are misunderstanding the argument.

In Landau's exposition the successor operation (denoted by the apostrophe or 'prime' symbol) is built into the Peano axioms; this operation, together with the use of the Induction Axiom, is then used (in Theorem 4) to prove that there exists a unique function $f$ on $\mathbb{N} \times \mathbb{N}$ with the properties that

1. $f(x,1)=x'$
2. $f(x,y')=f(x,y)'$

Once the proof of this theorem is complete, you can then introduce the notational convention that $x+y$ simply means $f(x,y)$.

What may be making Landau's exposition hard to follow is that, unlike what I wrote above, Landau does not bother messing around with writing $f(x,y)$; he introduces the notation $x+y$ at the same time that he states and proves the theorem. (That's why in the textbook it says "Theorem 4, and at the same time Definition 1".) That is a rhetorical technique that can be hard to decode if you're not used to it.

To the question of whether this is "typical of textbooks on real analysis these days": I think in general real analysis textbooks take one of two approaches. Some texts begin by defining (in approximately this order) group, field, ordered field, and complete ordered field (perhaps interposing Archimedian ordered field in between the third and fourth items of that list), then prove that any two complete ordered fields are isomorphic. This theorem then entitles them to say something like "From now on we will use the symbol $\mathbb{R}$ to refer to a complete ordered field; because any two complete ordered fields are isomorphic, we don't need to concern ourselves with what the real numbers 'are really'. All we need to know is that they are a complete ordered field, and ensure that everything else we say about them from now on can be derived from the field, order, and completeness axioms."

Of course it can be objected that:

• Proving uniqueness does not prove existence; for the latter, we would need to construct a real ordered field out of more basic elements.
• It seems perverse to develop an entire theory of the real numbers without ever saying what a real number is.

In response to these objections one might rebut: - Sure, but which set of basic elements? You can construct the reals by taking Dedkind cuts on the rationals, but then you have to ask "What are the rationals?" You can construct the rationals as equivalence classes of integers, but then you have to ask "What are the integers?" You can construct the integers as equivalence classes of natural numbers, but then you have to ask "What are the natural numbers?" You can construct the natural numbers out of ZF set theory, but then you have to ask "What are sets?" No matter how "deep" you go, eventually you have to stop and just stipulate some axioms about undefined terms. So why not just start at the level of "complete ordered field"?

Landau's "Preface for the Teacher" deals directly with these issues. In his context (1929 Germany) it's clear that the mainstream approach was to just start with the axioms of a complete ordered field and build up from there. In such an approach, addition does not get defined at all. It (and "natural number") are undefined terms; one just specifies what axioms it satisfies and moves on. Landau objected to this approach, feeling that the foundations of analysis should be pushed deeper back.