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 🙂