I wonder if there is a notion like the limit of formulas (and structures) because I believe it is important in describing countable structures (from finite structures). (For more detail, see [this paper](https://1drv.ms/u/s!AlRd0Qb77Om9bAn0KprNlCzxMWQ?e=q3Mtlj).) Now I give an excellent example illustrating it. The example is the so-called Quine atom. Let
$$
I\_n=\{I\_{n-1}\},\quad\text{for } n\geqslant1, \:\text{and } I\_0=G\_0.
$$
Where $G\_0\neq\{G\_0\}$. By repeatedly applying the axiom of extensionality, we get
\begin{align}
\phi\_n&\iff\exists !\:y\_n\in I\_n
\\
&\iff \exists !y\_n(y\_n=I\_{n-1})
\\
&\iff\exists !y\_n((\exists !y\_{n-1}\in y\_n)(y\_{n-1}\in I\_{n-1})\land (\exists !y\_{n-1}\in I\_{n-1})(y\_{n-1}\in y\_n))
\\
&\iff\exists !y\_n \exists !y\_{n-1}(y\_{n-1}\in y\_n\land y\_{n-1}\in I\_{n-1})
\\
&\iff\exists !y\_n \exists !y\_{n-1}(y\_{n-1}\in y\_n)\land \exists !y\_{n-1}(y\_{n-1}=I\_{n-2})
\\
&\iff\exists !y\_n \exists !y\_{n-1}(y\_{n-1}\in y\_n)\land\exists !y\_{n-1}((\exists !y\_{n-2}\in y\_{n-1})(y\_{n-2}\in I\_{n-2})\land(\exists !y\_{n-2}\in I\_{n-2})(y\_{n-2}\in y\_{n-1}))
\\
&\iff\exists !y\_n \exists !y\_{n-1}(y\_{n-1}\in y\_n)\land \exists !y\_{n-1}\exists !y\_{n-2}(y\_{n-2}\in y\_{n-1}\land y\_{n-2}\in I\_{n-2})
\\
&\quad\:\vdots
\\
&\iff \exists !y\_n \exists !y\_{n-1}(y\_{n-1}\in y\_n)\land \bigwedge\_{1\leqslant j\leqslant n-1}\exists !y\_j\exists !y\_{j-1}(y\_{j-1}\in y\_{j})\quad\quad\quad\quad\text{(1)}
\end{align}
Clearly, for any $n<\omega$, $I\_{n}\models \phi\_n$. This suggests that we can define the limit of formulas as follows.


**Edit:** As pointed out in @Goldstern's example, formulas for limit can not be too general. So we limit them to a $\aleph\_0$-categorical theory.


**Definition 1**: Suppose for any $n<\omega$, $\varphi\_{n}$ is in a $\aleph\_0$-categorical theory $T$ in $L$ (an infinitary language of $L\_{\omega\_1, \omega}$) and $M\_{n}\models \varphi\_{n}$. If for any $n$, there is a $N\_n$ that for any $k>N\_n,\:M\_{k}\models \varphi\_{n}$, then $\lim\_\limits{n\to\omega}\varphi\_{n}$ is a (unique) formula (up to equivalence) in $L$, and $\lim\_\limits{n\to\omega}M\_{n}$ is a (unique) $L$-structure. Let $\lim\_\limits{n\to\omega}\varphi\_{n}=\varphi\_{\omega}$ and $\lim\_\limits{n\to\omega}M\_{n}=I\_{\omega}$. Then $M\_{\omega}\models \varphi\_{\omega}$.


Also the follow axioms hold for the limit of formulas.


1. $\lim\_\limits{n\to\omega}(\phi\_{n}\land\varphi\_n)=\lim\_\limits{n\to\omega}\phi\_{n}\land\lim\_\limits{n\to\omega}\varphi\_n$
2. $\lim\_\limits{n\to\omega}\neg\phi\_{n}=\neg\lim\_\limits{n\to\omega}\phi\_{n}$
3. $\lim\_\limits{n\to\omega}\exists x\phi\_{n}=\exists x\lim\_\limits{n\to\omega}\phi\_{n}$
4. $\lim\_\limits{n\to\omega}\bigwedge\_\limits{1\leqslant j\leqslant n}\phi\_j=\bigwedge\_\limits{ n<\omega}\phi\_n$


We can prove from the above axioms that


5. $\lim\_\limits{n\to\omega}(M\_{n-1}\in M\_n)=\lim\_\limits{n\to\omega}M\_{n-1}\in \lim\_\limits{n\to\omega}M\_{n}$
6. $\lim\_\limits{n\to\omega}\exists x\_n(M\_{n-1}\in M\_n)=\exists\lim\_\limits{n\to\omega} x\_n(\lim\_\limits{n\to\omega}M\_{n-1}\in \lim\_\limits{n\to\omega}M\_{n})$


(For detailed proofs, see [this paper](https://1drv.ms/u/s!AlRd0Qb77Om9bAn0KprNlCzxMWQ?e=q3Mtlj).)


$\operatorname{Th}(I\_n)$ is $\aleph\_0$-categorical because any $I\_i$ can be mapped one-on-one to $I\_j$, and so is homogeneous. Since the language of set theory is finite relational, $\operatorname{Th}(I\_n)$ is $\aleph\_0$-categorical. By definition 1 and (1), we can see that $\lim\_\limits{n\to\omega}\phi\_{n}$ and $\lim\_\limits{n\to\omega}I\_{n}$ both are unique. Let $\lim\_\limits{n\to\omega}\phi\_{n}=\phi\_{\omega}$ and $\lim\_\limits{n\to\omega}I\_{n}=I\_{\omega}$. Then $I\_{\omega}\models \varphi\_{\omega}$. Furthermore
\begin{align}
\phi\_{\omega}&=\lim\_\limits{n\to\omega}\phi\_{n}
\\
&=\lim\_\limits{n\to\omega}\exists !y\_n \exists !y\_{n-1}(y\_{n-1}\in y\_n)\land \lim\_\limits{n\to\omega}\bigwedge\_{1\leqslant j\leqslant n-1}\exists !y\_j\exists !y\_{j-1}(y\_{j-1}\in y\_{j})
\\
&=\exists !I\_{\omega}(I\_{\omega}\in I\_{\omega})\land \bigwedge\_{n<\omega}\exists !y\_n\exists !y\_{n-1}(y\_{n-1}\in y\_{n})\quad\quad\quad(\lim\_\limits{n\to\omega}y\_n=I\_{\omega})
\end{align}


Thus $I\_{\omega}=\{I\_{\omega}\}$, i.e. $I\_{\omega}$ is a Quine atom.


Since the limit of formulas (for finite structures) can completely describe the Quine atom, I believe that it (will) play a significant role in the investigation of countable structures. I'd like to confirm that the above notion of limit of formulas and reasoning are not available in current model theory and so belong to a new field of research.


**Edit:** Next I will give two examples on how to apply the limit of formulas to studying some known results in model theory.


**Example 1:** There is arbitrary large number in nonstandard number theory.


Let $\phi\_n=\exists x\bigwedge\_\limits{m\leqslant n}(x>m)$ and $M\_n\models \phi\_n$. Since for any $k>n, \:x>k\to x>n, \:M\_k\models \phi\_n$. So the limit of $\phi\_n$ exists and
$$
\lim\_\limits{n\to\omega}\exists x\bigwedge\_\limits{m\leqslant n}(x>m)=\exists x\bigwedge\_\limits{n<\omega}(x>n)
$$
And there is a $M\models \exists x\bigwedge\_\limits{n<\omega}(x>n)$.


The second example shows that theory of DLO without endpoints is the limit of union of integers shifts.


**Example 2:** Suppose $T$ is the theory of DLO without endpoints and $M\_n=\bigcup\_\limits{1\leqslant j<n}(\Bbb{Z}+j/n)$. Then $\lim\_\limits{n\to\omega}\bigcup\_\limits{n<\omega}M\_n=\Bbb{Q}$ and $T = Th(\Bbb{Q})$.


Suppose $\varphi\_n,\phi\_n,\delta\_n$ are sentences specifying the properties of linear ordering, a dense subset and set without endpoints for $M\_n$. Then


\begin{align}
\varphi\_n &= (∀x,y,z∈M\_n)(x≤x∧(x≤y∧y≤x⇒x=y)∧(x≤y∧y≤z⇒x≤z))
\\
\phi\_n &= (∀x,y∈M\_n)(x<y⇒(∃z∈M\_{l})(l>n∧x<z<y))
\\
\delta\_n &= (∀x∈M\_n)((∃y∈M\_n)(y<x)∧(∃y∈M\_{n})(x<y))
\end{align}
For any $x,y∈M\_{n}\:(x<y)$, set $N\_{n}=2n$. Then $∀k>N\_{n},\:∃z∈M\_{k}$ that $x<z<y$, i.e. $M\_{k}\models φ\_{n}$. Since $ℤ\modelsϕ\_{n}∧δ\_{n},\:M\_{k}\models ϕ\_{n}∧φ\_{n}∧δ\_{n}$ and $T$ is $ℵ\_0$-categorical, $\lim\_\limits{n\to\omega}M\_n$ is unique. And
\begin{align}
\lim\_\limits{n\to\omega}ϕ\_{n} &=(∀x,y,z∈\lim\_\limits{n\to\omega}M\_n)(x≤x∧(x≤y∧y≤x⇒x=y)∧(x≤y∧y≤z⇒x≤z))
\\
\lim\_\limits{n\to\omega}\phi\_{n} &=(∀x,y∈\lim\_\limits{n\to\omega}M\_n)(x<y⇒(∃z∈\lim\_\limits{n\to\omega}M\_n)(x<z<y))
\\
\lim\_\limits{n\to\omega}δ\_{n} &=(∀x∈\lim\_\limits{n\to\omega}M\_n)((∃y∈\lim\_\limits{n\to\omega}M\_n)(y<x)∧(∃y∈\lim\_\limits{n\to\omega}M\_n)(x<y))
\end{align}
Since $\lim\_\limits{n\to\omega}ϕ\_{n},\lim\_\limits{n\to\omega}φ\_{n}, \lim\_\limits{n\to\omega}δ\_{n}$ are axioms of $Th(\Bbb{Q}),\: \lim\_\limits{n\to\omega}M\_n=\Bbb{Q}$ and $T = Th(\Bbb{Q})$.



Is there a concept of limit of formulas
I am not sure which set-theoretic axioms you want to use. Certainly not foundation, but I guess that at least the singleton axiom is allowed.


Consider $M\_0:= \{x\}$, where $x$ is any element satisfying $x\not=\{x\}$. $M\_{n+1}:=\{M\_n\}$.
Let $\varphi\_n$ be $\phi\_n \wedge \psi$, where $\phi\_n$ is your formula, and
$\psi$ says the unique element is not an element of itself. Then by your arguments the limit $M\_\omega$ must satisfy $M\_\omega = \{M\_\omega\}$, but it must also satisfy $\psi$, which is a contradiction.


So your axioms and rules lead to a proof of $x=\{x\}$ for all $x$.


It may be that I missed something. As a matter of fact, I am only guessing at what formal language you are using.


