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Here we show that the A_TM problem is undecidable and recognizable, which is asking if there is a decider for whether an arbitrary Turing Machine accepts an arbitrary input. The proof is by contradiction and diagonalization.
What is a Turing Machine? It is a state machine that has a set of states, input, tape alphabet, a start state, exactly one accept state, and exactly one reject state. See • Turing Machines - what... for more details.
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I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.