Imagine you have an code which does #2. Now give it itself as an argument. What is the result? If it emits 0 then it's behaving according to #1, then because we

Imagine you have an code which does #2. Now give it itself as an argument. What is the result? If it emits 0 then it's behaving according to #1, then because we assumed it satisfies #2 it must emit 1, a contradiction. Similarly it can't emit 1. Thus it is impossible to have an algorithm which solves #2 for all inputs, because it must fail on at least one input (itself)