Can a solvable problem be encoded in a recursively enumerable language? Imagine I have a turing machine that can decide on a specific problem using a language. My question is that that problem (that can be decided by a TM M, with language L) can be encoded in a new language that is recursive enumerable.

If yes, this would mean that every solvable problem can encoded in recursive language; if not, a problem can be encoded in a recursive and/or recursive enumerable L and the problem could become semi-decidable for a different language.

Any clarification on this?
Almost immediately from definitions, one has:


A solvable (decidable) problem can be encoded as (not in) a recursive language. 
A semi-decidable problem can be encoded as (not in) a recursively enumerable language.