# Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.

Setting

For any language $\mathcal L$, two $\mathcal L$-structures $\mathcal M$ and $\mathcal N$ are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.

Attempt

($\Rightarrow$) Given $\mathcal M\equiv \mathcal N$, so $\mathcal M\models \phi \iff \mathcal N \models \phi$ for every $\mathcal L$-sentence $\phi$. Now suppose there is finite sublanguage of $\mathcal L$ where $\mathcal M\not\equiv \mathcal N$, then it follows that we can find some $\mathcal L$-sentence $\phi$ where $\mathcal M\models \phi$ but $\mathcal N \not\models \phi$, contradicting the assumption that $\mathcal M\models \phi \iff \mathcal N \models \phi$ for every $\mathcal L$-sentence $\phi$.

($\Leftarrow$) Now suppose $\mathcal M\equiv \mathcal N$ in every finite sublanguage, then it follows by compactness $\mathcal M\equiv \mathcal N$.

Problem

In the $\Rightarrow$ direction, I am not confident that I can use compactness. Since each finite sublanguage may not necessarily generate a finite set of sentences, which compactness theorem requires.

• Suppose $\mathcal M \not \equiv \mathcal N$. Then there is a formula $\phi$ such that $\mathcal M \models \phi$ and $\mathcal N \models \lnot \phi$. Now use the hint. – Levon Haykazyan Apr 22 '15 at 10:24