Language of Recursively enumerable languages

Question

Is the set of all recursively enumerable languages recursively enumerable. In other words is the following language L recursively enumerable. L={$\langle M \rangle$| Where M is a TM}

Can anybody help me in proving this?

Answer 1

Depending on your definitions of the notation of a TM, your set could even be regular. If transitions that use states or symbols that are not in the TM's state set/tape alphabet are allowed and simply ignored, and maybe some other details (fixed start state for all TM), the syntax for a correct TM should be regular. It is definitely recursively enumerable and even context-sensitive, even if you are restrictive on what a valid specification is (no inexistent states/smybols etc.). Not much needs to be checked.

However, the resulting language should not be called the "Language of Recursively enumerable languages" as you do in your title. It is merely the language of all valid Turing Machine specifications.