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That won't wash. The entropy of a system does not depend one tiny bit
on "the INFORMATION we have about" it! If you believe that it does,
you believe in magic. The entropy of a physical system (apart from a
question about its zero point) is a function only of its thermodynamic
coordinates, and those depend not at all on human knowledge. My entire
problem with teaching this particular pernicious factoid is that it
tends to mystify physics. Students hear it and quite justifiably think
that physicists believe in magic.