Assume that $\mathcal{C}=\{M_i:i\in I\}$ is an infinite collection of different finite $\mathcal{L}$-structures in a first-order language $\mathcal{L}$. The question is:
- What kind of infinite "limits" can we produce from the class $\mathcal{C}$?
- What kind of "transference principles" are valid for the different limits that we can take.
- What advantages could exist by working in the limit $\mathcal{M}$ instead of working directly on the class $\mathcal{C}$.
Perhaps the prototypical example is the ultraproduct construction, on which the transference principle is Łoś's Theorem and has the advantage of, among other things, being an infinite structure and also being equipped with some notion of measure for definable sets.
However, I have heard also about profinite structures (inverse limits of finite structures), and about Fraïssé limits, among others. But I am not sure what are their main properties, or if there exist some sort of transference principle (perhaps only for some kind of formulas). I would appreciate if anyone can give any information or reference to look at.
The ultimate answer would be a complete classification of the possible limits together with their properties, but of course I understand that such a thing could only exist in the platonic world. Thus, any partial answer will be appreciated as well.
Since the questions is fairly broad, the answer below may be a bit trivial, but there are more interesting examples given afterwards with proper reference.
Let's take a converging sequence $(M_n) \to \mathcal{M}$ (all that follows also applies to random sequences). To precisely define what convergence mean, let's define the metric $$d_{FO}(M_1,M_2) = \begin{cases} 2^{-\min(q(\varphi))} & \mbox{for } M_1 \models \varphi \wedge M_2 \not\models \varphi\\ 0 & \mbox{otherwise}\\ \end{cases}$$ (where $q(\varphi)$ is the quantifier depth of the formula). So this metric depends on the smallest formula separating two structures. Let's assume we have properly defined our space so that we have as usual that a sequence is convergent iff it is Cauchy. Two structures are elementarily equivalent if they are at distance $0$. Since finite structures are categorical, the equivalence class of all finite structures are singletons, but the equivalence classes of infinite structures may be infinite. Call an infinite structure an $*$-limit if it can be obtained as the limit of a converging sequence of finite structures.
Now for the consequences of these definitions:
(1). All $*$-limits have the finite model property (FMP). However characterizing theories with the FMP is a hard task.
(2). Similarly, by definition you have $\mathcal{M} \models \varphi$ implies that there exists an $i$ such that for all $n>i$, $M_n \models \varphi$. Getting an estimate on $i$ depends on the rate of convergence.
(3). I assume you mean what is the advantage to work on $\mathcal{M}$ rather than $(M_n)$. Well, if you don't have estimates on the rate of convergence, if you have $M_i \models \varphi$, you don't know if it isn't an anomaly because you took a too small structure. So some tests might work better on the limit since it may be free of such artifacts. However all non-first-order parameters (e.g. connectedness) are not continuous with the limit, so it may not be accurate to use for all estimations.
An interesting example (or rather a counter-example) of the link between Fraïssé limits and a class of finite structures is the class of triangle-free graphs. The limit of the sequence $(T_n)$, where $T_n$ is the uniform distribution on all triangle-free graphs on $n$ vertices, is $\mathcal{B}$, the bipartite equivalent to the Rado graph. This happens because almost all triangle-free graphs are bipartite. However, the Fraïssé limit of the triangle-free graphs is not bipartite, so there is no transference principle at all. This contrasts sharply with the class of all graphs and the Rado graph. See the details, proofs and wonders in [0].
For a classification of all possible limits... there are a lot of them, right? However there are some interesting results of more restricted scope. Way too much to give an overwiew here, but we can have two interesting examples. There's a classification of reducts of the Rado graph [1], or Ove Ahlman's work [2], which precisely study the class of structures that can be obtained as limits of sequences (with some additional hypothesis).
[0] Kolaitis, Promel, Rothschild, "$K_{l+1}$-free graphs: asymptotic structure and a 0-1 law", Trans. Amer. Math. Soc., 303 (1987).
[1] Simon Thomas, "Reducts of the Random Graph", J. Symbolic Logic 56 (1991).
[2] Ove Ahlman, "Simple structures axiomatized by almost sure theories", Annals of Pure and Applied Logic 167 (2016).