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Trying to answer the many questions you have here to the best of my ability: what is R(M)? They say it is representation of turing machine but what is it exactly? Is it tuples of turing machine? It is some representation of some Turing Machine $M$. The exact format of the representation is not important for this proof, just that it unambiguously specifies ...


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This statement is incorrect. If $\overline{A_{TM}}$ is co-recognizable, then we have a TM that recognizes the set of words not accepted by some TM $M$. The fact that $\overline{A_{TM}}$ is co-recognizable means that there is a TM $T$ that recognizes all words not in $\overline{A_{TM}}$. Equivalently, $T$ recognizes all words in $\overline{\overline{A_{TM}}}...


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