In some books, they seem to implicitly say that $S(x) = x+1$ holds always in Peano arithmetic. But does it really hold in all cases, even in non-standard ones? The standard model of course satisfies this, but non-standard ones don't seem obvious.
Also, in Peano arithmetic, some books also say that $0<1$ all the times. Is this true for all non-standard models of Peano arithmetic?
If all these are true, is this solely due to axiom schema of induction?
Note that $1$ is a shorthand for $S(0)$. Our language only contains the symbols $0,S,+,\cdot$ and no symbol for $1$.
The axioms for addition tell us that $x+0=x$ and $S(x+y)=x+S(y)$. Therefore we have:
$$S(x)=S(x+0)=x+S(0)=x+1.$$
As you can see I didn't use induction, at all.