I'm reading The Number Systems — Foundations of Algebra and Analysis, second edition, by Solomon Feferman. I have a doubt regarding the statement of theorem 4.22, chapter 4.
Theorem:
--- There exists an ordered integral domain (I, +, . , <, 0, 1) which
(i) contains (P, +, . , <, 1) as a subsystem, and satisfies
(ii) for any x ∈ I either x ∈ P or x = 0 or -x ∈ P. ---
My question is whether it is correct to say that (P, +, . , <, 1) is a subsystem of (I, +, . , <, 0, 1), since 0 is not a specific element of P. In fact, 0 is not an element of P (set of positive integers).
In the definition of subsystem given in Chapter 2, each subsystem has the same specified elements as the system that contains it.
I ask for help.