Trivial Extensional Language Property

languages

Definition

Trivial Extensional

Fix an alphabet $\Sigma$. An extensional family $\mathcal{E}\subseteq {\mathrm{RE}}_{\Sigma }$ is called trivial if either $\mathcal{E}=\varnothing$ or $\mathcal{E}={\mathrm{RE}}_{\Sigma }$ (all recursively enumerable languages over $\Sigma$).

Examples:

• ${\mathcal{E}}_1=\{L\mid L\text{ is semi-decidable}\}={\mathrm{RE}}_{\Sigma }$ (trivial)
• ${\mathcal{E}}_2=\varnothing$ (trivial)