A word w is a member of a language L iff it is a member of the language L, ie. the condition of w’s membership to L is that it be a member of L. This is the definition of the language.
Without ahead of time specifically declaring a list of words which are in L–that would then be in L by definition–is this language decidable? Or, do we end up (as suspected), for any machine that decides L, and on any input, in an infinite loop due to the self-reference: we go to check if w is in L and then see that we need to check if w is in L and then see that we need to check if w is in L…ad inf and this language is undecidable?
PS. Is there a name for a language that refers to itself in its definition?