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Turing Machines
In 1928, David Hilbert, one of the most influential mathematicians of his time, asked whether it is possible to create an algorithm that could determine the correctness of a mathematical statement. This was called the "decision problem," or "Entscheidungsproblem" in Hilbert's native German.
At any point, feel free to !pause, step the machine !forwards or !backwards one instruction at a time, or !restart the program from the beginning. It seems like a silly example, the kind of answer a cheeky high school student might try to get away with, but it is a legitimate counterexample to the idea that the halting problem can be solved. Everyone who let me watch them read this post in real-time over a video call and gave me feedback: Jaga Santagostino, Robert Aboukhalil, Tarun Verghis, Tyler Sparks.
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