# Check if language is decidable

I would like to determine if the following language is decidable or not. L = { w $$\in$$ $$\Sigma^*$$ | $$T(M_w)$$ is recognized by a Turing machine with at most 42 states}.

I know that every finite language is decidable, but I am not sure if this has anything to do with this particular problem.

Every help is appreciated, I am a little lost.

Thank you

• What is T(M)? Is it the language generated from the turing machine M? May 28, 2020 at 14:23

Define C = { $$L | L$$ is decidable by a turing machine with at most 42 states }
Notice that there is a finite amount of such languages, as there are a finite amount of turing machines with 42 states. for that reason, we have $$C\ne\emptyset, \sum^*$$
By Rice's theorem, the language $$S=$$ { $$w | L(M_w) \in C$$ } is undecidable. This language is precisely the language in question