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Turing's topological proof that every written alphabet is finite (2010)
Recently one of my colleagues was reading Alan Turing’s groundbreaking 1936 article “On Computable Numbers with an Application to the Entscheidungsproblem.” This is the article in…
If these sets are restricted to be measurable, we can define the “distance” between two symbols as the cost of transforming one symbol into the other if the cost of moving unit area of printer’s ink unit distance is unity, and there is an infinite supply of ink at , . We could have dropped the compactness assumption, but if we did then the Hausdorff metric becomes a pseudometric and things get a little messier.] Thus we can use the identical argument, but now a symbol is a compact subset of , to conclude that there are only finitely many color pictures on a given canvas.
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