In his book Number Systems and the Foundations of Analysis, Elliot Mendelson first defines a Peano system (p. 53). He then goes on to prove that two arbitrary Peano systems are isomorphic — they are essentially the same (p. 80). He then makes a basic existence assumption — there exists a Peano system (p. 83).
In his book The Real Numbers and Real Analysis, Ethan D. Bloch states Axiom 1.2.1 (Peano Postulates) (p. 3). Axiom 1.2.1 states the existence of a Peano system, namely $(\mathbb{N}, s, 1)$. Then, in Exercise 1.2.8 (p. 11), he mentions that two (arbitrary) Peano systems are isomorphic. So, I think Bloch’s approach is (partly) equivalent to Mendelson’s. (Bloch never defines what Peano systems are.)
Bloch then defines the set of natural numbers (Definition 1.2.2, p. 3) as "the set the existence of which is given in the Peano Postulates." It implies that the members of a Peano system are called natural numbers. (It reminds me of how we define vectors as elements of a vector space.) So, in his treatment of natural numbers, it seems that natural number isn’t a primitive notion.
On the other hand, Landau, in his book Foundations of Analysis, doesn’t try to define natural numbers. Instead, he takes it as an undefined term, which, I think, is how an axiomatic treatment works.
If I wanted to define what natural numbers are, would the following sequence be logically accurate?
- State the definition of a Peano system.
- Prove that two arbitrary Peano systems are isomorphic.
- Assume that there exists a Peano system.
- Define natural numbers as elements of the underlying set of that Peano system.
