# Finite languages $L\in RE$

I want to check if I understood it in the right way. In some example where $$L\in RE$$ the explanation deal with 2 cases: 1st when $$L$$ finite and 2nd when $$L$$ infinite. In the second case $$L\in R$$, isn't? Is it possible somehow that $$L\in RE\setminus R$$ be finite?

• The question is unclear to me. For example, I don't understand what you mean by "The second case is 𝐿∈𝑅, isn't?". Regarding the last question, every finite language is regular and thus decidable. Jan 15 at 19:49
• Thanks @BaderAbuRadi, I edited the question is it clear now? Jan 15 at 19:58
• Sorry, it is still unclear. Are you asking: 1) whether every infinite recognizable language $L$ is in $R$? 2) whether there are finite languages in $RE\setminus R$? Jan 15 at 20:22
• Whether there are finite languages in 𝑅𝐸∖𝑅? Jan 15 at 20:25

The answer is no. Every finite language is regular, and thus decidable. Therefore the existence of a finite language $$L$$ in $$\text{RE} \setminus \text{R}$$ is impossible. However, note that there languages $$L$$ in $$\text{RE}\setminus \text{R}$$ (e.g., $$Halt_{TM}$$), and by what we have seen previously, such languages have to be infinite.