Let $I$ be the set of all recursively enumerable languages over an alphabet $\Sigma$.Let $$S_\alpha=\{i\in I : \alpha\in i\}$$ for all $\alpha\in\Sigma^*$.Then $$E=\{S_\alpha:\alpha\in K\subseteq\Sigma^*\}$$ can be extended to an ultrafilter $\mathscr{U}$ since $E$ has the finite intersection property:$$\{{\alpha_1},\dots,{\alpha_n}\}\in S_{\alpha_1}\cap\dots\cap S_{\alpha_n}\text { and }\{{\alpha_1},\dots,{\alpha_n}\}\in I \text{ since it's a finite language}$$We can define $L=\{\alpha:\{i\in I:\alpha\in i\}\in \mathscr{U}\}$.What is the relationship between $L$ and $K$?
Formal languages induced by ultrafilters
fl.formal languages