## About a “modification” of the diagonal language $\{w_i \mid \text{Every turing machine } M_1 \ldots M_i \text{ reject } w_i\}$

I have given the seeming modification of the diagonal language $$\{w_i \mid \text{Every turing machine } M_1 \ldots M_i \text{ rejects } w_i\}$$, yet I can’t prove that it is undecidable.

My thoughts so far:

• This language is intuitively undecidable, but it might trick you into thinking that it is, and it is in fact decidable: At last there exists an $$i$$ from which $$M_i = M^*$$ where $$M^*$$ rejects every word $$w$$

• I can’t directly reduce this to the diagonal language

• I can’t build a halting problem solving machine on this if I can’t define my enumeration of $$w_i$$ and $$M_i$$ freely, which I can’t (and is that even right?).

A nice tip would be helpful. Thanks 🙂