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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.

Paulo Argolo
  • 4,210
  • You may just drop $0$ in the second system, and define $0$ using $+$ in the obvious way, so (ii) became $x\in P$ or $\forall y\in I, x+y=y$ or $\exists y\in P, \forall z\in P, x+y+z=z$. I would argue though, you add something to a system (rather its signature) doesn't change the fact that it's still an example of the systems of an old type. – Just a user May 23 '22 at 12:42
  • Well... I keep thinking that (P, +, . , < , 1) is not a subsystem of (I, +, . , < , 0, 1). This is what the author's own definition of subsystem tells me. – Paulo Argolo May 23 '22 at 17:31

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