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The author introduces a "harmony versus dissidence" way of looking at the Lebesgue measure as far as the construction of non-measurable subsets of the real line is concerned, and then he illustrates this idea using examples of different reduced automata that faithfully imitate a certain incompletely specified automaton. This is in order to show that ambiguity always and everywhere begets confusion in mathematics. Then he suggests using the first digit phenomenon work that he has done that the relationship between the real line's topology and its Lebesgue measure is very delicate so that the usual way to construct a non-measurable subset of the reals by wrapping the real line around a unit circle is leading to confusion in such areas because the finding of new axioms for set theory to solve the continuum problem is jeopardized by this confusion.
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