The book I am earning from states that the theory of a $\mathcal{V}$-structure, $M$, is the set of all $\mathcal{V}$-sentences that are modeled by $M$. Then it goes on to say that that any theory of a $\mathcal{V}$-structure is complete. So lets look at the structure $NT= (\mathbb{N} | +, \cdot,1)$.

My book says that the axioms of NT are:

$\forall x \lnot(x+1=1)$

$\forall x \forall y (x+1 = y+1 \implies x=y)$

$\forall x \forall y (x+y = y+x)$

$\forall x \forall y (x + (y+1)=(x+y)+1)$

$\forall x \forall y (x \cdot y = y \cdot x)$

$\forall x (x \cdot 1 = x)$

$\forall x \forall y (x \cdot (y+1) = (x \cdot y)+x)$

$\forall^1 S(S(1) \land \forall x(S(x) \implies S(x+1)))\implies \forall x S(x)$

This last axiom is technically a second-order sentence, but I have read that it can be written in a first order language as an axiom schema. So these sentences, by our choice, are in $Th(NT)$. But Godel showed that, only assuming these axioms, there are some true sentences which cannot be derived. So won't these sentences and their negations be missing from $Th(NT)$? I don't understand how my book can claim that $Th(NT)$ is complete.