My teacher presented Gödel's first incompleteness theorem as follows:
(1) First Incompleteness theorem (Gödel [Rosser]): Every [$\omega$-]consistent and reasonably expressive system of arithmetic contains sentences that are neither provable nor refutable
(2) First Incompleteness Theorem (with shades of Tarski): Every sound and reasonably expressive system of arithmetic contains true, but unprovable sentences.
This is still a bit vague, so here the formalisation of a "system of arithmetic":
Def. A system $\Sigma$ is a set containing the following components:
$\mathcal{E} \ldots $ expressions of $\Sigma$
$\mathcal{S} \ldots $ sentences of $\Sigma$; $\mathcal{S} \subseteq \mathcal{E} $
$\mathcal{T} \ldots $ true sentences of $\Sigma$; $\mathcal{T} \subseteq \mathcal{S}$
$\mathcal{P} \ldots $ provable sentences of $\Sigma$; $\mathcal{P} \subseteq \mathcal{S}$
$\mathcal{R} \ldots $ refutable sentences of $\Sigma$; $\mathcal{R} \subseteq \mathcal{S}$
$\mathcal{H} \ldots $ predicates of $\Sigma$; $\mathcal{H} \subseteq \mathcal{E}$
function $\Phi: \mathcal E \times \mathcal H \mapsto \mathcal E$: If $E \in \mathcal H$ then $\Phi(E, n) = E(n) \in \mathcal S$
Def. Soundness: A system is sound iff all provable sentences are true ($\mathcal P \subseteq \mathcal T$) and no refutable sentence is true ($\mathcal R \cap \mathcal T = \emptyset$).
Def. Completeness: A system is complete iff all true sentences are provable ($\mathcal T \subseteq \mathcal P$).
What I am struggling with is how this formalisation translates to FOL and the notion of first-order theories. In particular, how does "true sentence" translate to FOL? Does it mean tautological truth (in all structures)? Truth in a first-order theory (e.g. Peano Arithmetic)? Truth in just some structure (e.g. the standard model for arithmetic)?
I am also interested in how this relates to Gödel's completeness theorem of FOL. For example, if "true sentence" in the above system would mean "tautological truth of FOL", then (2) would contradict Gödel's completeness theorem of FOL. Right?