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.) 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. (It could be extended to the theory of a countably saturated structure though.)

**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.

- $\lim_\limits{n\to\omega}(\phi_{n}\land\varphi_n)=\lim_\limits{n\to\omega}\phi_{n}\land\lim_\limits{n\to\omega}\varphi_n$
- $\lim_\limits{n\to\omega}\neg\phi_{n}=\neg\lim_\limits{n\to\omega}\phi_{n}$
- $\lim_\limits{n\to\omega}\exists x\phi_{n}=\exists x\lim_\limits{n\to\omega}\phi_{n}$
- $\lim_\limits{n\to\omega}\bigwedge_\limits{1\leqslant j\leqslant n}\phi_j=\bigwedge_\limits{ n<\omega}\phi_n$

We can prove from the above axioms that

$\lim_\limits{n\to\omega}(M_{n-1}\in M_n)=\lim_\limits{n\to\omega}M_{n-1}\in \lim_\limits{n\to\omega}M_{n}$

$\lim_\limits{n\to\omega}\exists x_n(M_{n-1}\in M_n)=\exists\lim_\limits{n\to\omega} x_n(\lim_\limits{n\to\omega}M_{n-1}\in \lim_\limits{n\to\omega}M_{n})$

(For detailed proofs, see this paper.)

$\operatorname{Th}(I_n)$ 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 $T = Th(\Bbb{Q})$.

peculiarthat it is almost impossible to tell what you are saying. $\endgroup$ – Andrés E. Caicedo Sep 11 at 0:21