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In metalogic: Syntax and semantics
… (sentences or meaningful expressions), applicable mechanically, in the sense that a machine could check whether a candidate satisfies the requirements. This specification usually contains three parts: (1) a list of primitive symbols (basic units) given mechanically, (2) certain combinations of these symbols, singled out mechanically as forming the simple (atomic)…
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In metalogic: Discoveries about formal mathematical systems
…intuitive concept of what a mechanical computing procedure is. As a result of the development of recursion theory, it is now possible to prove not only that certain classes of problems are mechanically solvable (which could be done without the theory) but also that certain others are mechanically unsolvable (or…
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In metalogic: The undecidability theorem and reduction classes
Once the concept of mechanical procedure was crystallized, it was relatively easy to find absolutely unsolvable problems—e.g., the halting problem, which asks for each Turing machine the question of whether it will ever stop, beginning with a blank tape. In other words, each Turing machine operates in a predetermined…
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