I am reading A Friendly Introduction to Mathematical Logic by Leary and Kristansen. The authors show that the logical axioms are valid. Since the logical axioms are independent of the language, they are true in any $\mathcal{L}$-structure and any first order language $\mathcal{L}$.
This gets confusing when it comes to nonlogical axioms. Peano Axioms $N$ (see Example 2.8.3) are nonlogical axioms in the language of number theory $\mathcal{L}_{NT}$. Here's my question: If $\mathfrak{N}$ is the standard structure of the number theory, is it correct to say that $\mathfrak{N}\models N$? Is it simply true by definition? Or do we accept nonlogical axioms to be true in any structure?
Likewise, if we are in the language of set theory, do we call the structure $\mathfrak{S}$ to be "standard" where ZF is accepted as nonlogical axioms?
Also, the authors prove that there is no way that a set of $\mathcal{L}_{NT}$ sentence can ever characterise the standard structure, i.e., even if there was some other structure such that $\mathfrak{A}$ where the Peano axioms $N$ was true then it would not hold that $\mathfrak{A}\simeq\mathfrak{N}$.
Best Answer
Exactly; we assume that Peano Axioms are true in the standard model.
And no; non-logical axioms are not valid, i.e. true in every structure. The axiom $¬∃x(0=s(x))$ does not hold in a domain with only one object.
And yes, there are non-standard models of arithmetic, i.e. structures satisfying first-order Peano arithmetic that contains non-standard numbers.
Not every collection of non-logical axioms have a "standard" model; see Group theory: every structure satisfying the group axioms is called a group, but there are many "varieties".
In the same way, there are many different structures called vector spaces: everyone satisfies the axioms for vector space.