What's the "right notion of sameness" for models?

Question

What constitute "different models" from the perspective of model theory?

The natural numbers $\mathbb{N}$ are a model of Peano arithmetic, here denoted $T_{PA}$.

I'm going to define $\mathbb{N}$ for the purposes of this question as the familiar construction $\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}, \cdots$ where $\in$ is $\le_{\mathbb{N}}$.

It seems pretty clear that $\mathbb{N} \models T_{PA}$, in fact, scraping away some of the detail of $\mathbb{N}$ is the point.

I can define a different model, $\mathbb{P} \subsetneq \mathbb{N} \times \mathbb{N}$, where each element of $\mathbb{P}$ is an ordered pair of the same number (in the set construction above).

$$ \mathbb{P} := \{ (\emptyset, \emptyset)\; , \;\; (\{\emptyset\}, \{\emptyset\})\; , \;\; (\{\emptyset, \{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}) \; , \;\; \cdots \} $$

With $\le_\mathbb{P}, +_\mathbb{P}, *_\mathbb{P}$ defined pointwise using $\le_\mathbb{N}, +_\mathbb{N}, \text{and} \;\; *_\mathbb{N}$, respectively

$(\mathbb{N}, \le_\mathbb{N}, +_\mathbb{N}, *_\mathbb{N})$ and $(\mathbb{P}, \le_\mathbb{P}, +_\mathbb{P}, *_\mathbb{P})$ are not literally the same, but they also aren't different in an interesting way.

To use an analogy from programming, it seems like models are particular programs written in some language $L$ and a theory $T$ is like an API, giving you signatures for functions and predicates as well as contracts that the implementation is required to obey. In the case above, $L$ seems to be a not-entirely-nailed-down set theory ... whose existence we are using to "breathe life into" our model and give it an existence independent of our theory.

Taking a step back from $L$ as set theory and just thinking of it like a generic programming language ... there's no obvious notion of sameness to me that's coarser than "textually equal"/"literally the same" and finer than "every model that models the same theory is equal".

What is the correct way of thinking about model sameness/equality?

Answer 1

Generally, two structures (in the sense of first-order logic - that is, a set together with some labelled relations, functions, and constants) are considered "the same" if and only if they are isomorphic:

If $\Sigma$ is a first-order language, then two $\Sigma$-structures $\mathcal{A},\mathcal{B}$ are isomorphic if there is a bijection $i$ between their domains which "preserves the $\Sigma$-structure" - e.g. for each unary function symbol $f$ in $\Sigma$ we need to have $i(f^\mathcal{A}(x))=f^\mathcal{B}(i(x))$ for every $x$ in the domain of $\mathcal{A}$.

This notion is - as you expected - much finer than elementary equivalence (= "have the same theory"), and much coarser than "textual equality" as you call it (in particular, it identifies the two versions of the naturals you describe).