I would like to prove the undecidability of the following language $L$, closely related to the Halting problem: $$L:=\{w \mid M_w \text{ accepts a word of size }|w|\}.$$
It is obvious for unary encodings, but for encodings that have several machines encoded by words of the same size, I don't see how I can prove this.
Here, $w$ is any Gödel Numbering of Turing machines. I.e. an effective recursive enumeration of all Turing machines. So $M_w$ is the Turing machine encoded by $w$.
Rice's Theorem does not apply directly. Maybe it can help, but it is neither obvious nor straight forward.
