Here is what I want to prove
$\Sigma$ a set of sentences such that there are arbitrarily large finite models in which $\Sigma$ is true. Prove that $\Sigma$ is true in some infinite model.
Here is my attempt:
Let $A$ be the domain of model $\mathfrak{A}$. Let $|A|=n$ be the size of $\mathfrak{A}$. If $n=0$, then $A=\emptyset$ and we have no variable assignment to talk about. Given $n$, consider sentence $\mathcal{T}$ such that \begin{align*} &\mathcal{T}:=\forall x_1\forall x_2\dots\forall x_n(\wedge_{j=1}^{n}\wedge_{i=1}^{n}(x_i\neq x_j))\text{ }\text{ when $n>1$}\\ &\mathcal{T}:=\forall x_1 x_1=x_1\text{ }\text{ when $n=1$} \end{align*} where $\wedge_{j=1}^{n+1}\wedge_{i=1}^{n}(x_i\neq x_j)=(x_1\neq x_2)\wedge (x_1\neq x_3)\wedge\dots\wedge (x_1\neq x_{n})\wedge (x_2\neq x_1)\wedge\dots$
If $n=1$, then any model with a smaller size is empty. And for any model with at least size $n$, since $x_1=x_1$ is true under any variable assignments, hence $\mathcal{T}$ is true in these models.
For $n>1$, if size of $\mathfrak{B}$ is less than $n$, then there must be at least one pair of identical elements in $\{b_1,\dots, b_{n}\}\subseteq B$ by Pigeonhole principle. Hence for any variable assignment $s$, if $s'\equiv s$ except $s'(x_i)=b_i$ $\forall 1\leq i\leq n$, then $s'(x_{i})=b_i= b_j=s'(x_{j})$ for some $1\leq i\neq j\leq n$. Hence $\mathcal{T}$ is never true in these model. If size of $\mathfrak{B}$ is at least $n$, then we can choose $\{b_1,\dots, b_{n}\}\subseteq B$ be pairwise distinct so that for any variable assignment $s$, if $s'\equiv s$ except $s'(x_i)=b_i$ $\forall 1\leq i\leq n$, then $s'(x_{i})=b_i\neq b_j=s'(x_{j})$ for all $1\leq i\neq j\leq n$, i.e. $\mathcal{T}$ is true under $s$ in $\mathfrak{B}$.
Now for each $n$, we get a sentence $\mathcal{T}$ that is true in all and only those models that have size $\geq n$. Let $\Sigma$ be a set of $\mathcal{T}$ for all $n$. Hence any finite subset of $\Sigma$ is true under some infinite model, so satisfiable. By Compactness, we know if every finite subset of a set of sentences is satisfiable then, the whole set is satisfiable, so we done.
My questions: Since $\Sigma$ is true under some arbitrarily large finite models, so I assume that for each $n$, every sentence in it is true in all model with size greater than $n$. However, I specifically choose my sentences to construct the $\Sigma$ instead of consider case in general. So am I wrong? And what will be a sentence look like in general such that given $n$, it is true in all and only those models that have size $\geq n$. Thanks in advance.
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Thanks @Asaf Karagila for pointing out the mistakes on the quantifier. $\mathcal{T}$ should be $\exists x_1\exists x_2\dots\exists x_n(\wedge_{j=1}^{n}\wedge_{i=1}^{n}(x_i\neq x_j))\text{ }\text{ when $n>1$}$. And the rest follow. And as mentioned by @Alex Kruckman, my question is a duplicate of a question asked by @MrTopology. I follow his idea to finish the problem in general case
Now for each $n$, we get a sentence $\mathcal{T}$ that is true in all and only those models that have size $\geq n$. Let $\Sigma^*$ be collection of $\mathcal{T}$ for all $n$. Hence $\Sigma^*$ is true in infinite model. Let $\overline{\Sigma}=\Sigma^*\bigcup\Sigma$. As $\Sigma$ is a set of sentences such that there are arbitrarily large finite models in which it is true, any finite subset of $\overline{\Sigma}$ is true under some finite model, so satisfiable. By Compactness, we know if every finite subset of a set of sentences is satisfiable then, the whole set is satisfiable, so $\overline{\Sigma}$ satisfiable. Since $\Sigma^*$ is true only in some infinite model under construction, hence $\overline{\Sigma}$ is true in some infinite model, then so is $\Sigma$