Confusion over the definition of "model"

We agree that an interpretation is a model of a formula $\varphi$ if the formula is TRUE in that interpretation (i.e. if the interpretation satisfies the formula).

The same for a set $\Gamma$ of formulas.

Consider now a theory $T$ with the collection $\Gamma_T$ of its axioms.

"A model of the theory is an interpretation that satisfies all the axioms of the theory."

So far, nothing new : the collection of axioms of the theory $T$ is a set of formulas. Thus, a model of the theory is an interpretation that satisfies all those formulas.

Consider for example first order arithmetic, i.e. the f-o version of Peano axioms.

We can call the collection of f-o Peano axioms with $\mathsf {PA}$.

We can prove from it the usual arithmetical laws (or theorems), like e.g. : $1+1=2$.

In symbols, we have : $\mathsf {PA} \vdash (1+1=2)$.

By soundness of the predicate calculus, we have : $\mathsf {PA} \vDash (1+1=2)$.

And again, this is consistent with the above definitions :

the arithemetical theorem $1+1=2$ is a logical consequence of the (first-order) arithmetical axioms, i.e. it is TRUE in every interpretation that satisfies the collection $\mathsf {PA}$ of arithmetical axioms.

In the answer to your previous post we have seen that an interpretation for propositional logic is :

an assignment $v : \text{At} \to \{ \text T, \text F \}$ such that, e.g. $v(p_0)= \text T, v(p_1)= \text F$, etc.

In the case of first-order language an interpretation needs a domain of "objects", like e.g. the set $\mathbb N$ of natural numbers.

Of course, different FOL theories need different domains, while in propositional logic we have only one domain: the boolean one $\{ \text T, \text F \}$.

And then we have to specify how to interpret the new elements of the language (in addition to the connectives) : quantifiers, individual constants, variables, predicate and function symbols.

Having done this, we have to complete our semantics for FOL with the definition of :

satisfaction relation, model, valid formula, logical consequence.