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Although in retrospect others (Bernard Bolzano, Richard Dedekind) can be viewed as precursors, set theory was largely the creation of a single individual, Georg Cantor, beginning in the 1870s, and his key work (Cantor, 1915) remains highly readable to this day. He launched the field with two results on questions with ancient roots. Pythagoreans noted that if the lengths of otherwise similar strings are in the ratio 2:1, the shorter sounds an octave higher. Why? Because it vibrates twice as quickly. In modern mathematical language, if the graph of the displacement of the center of the string with time approximates y ¼ cos x for the longer, it will approximate y ¼ cos 2x for the shorter. No real string vibrates so simply, and a better approximation for the long string would be y ¼ a1 cos þ a2 cos 2x; with the amplitude a1 of the “fundamental” much larger than the amplitude a2 of the “overtone.” By the eighteenth century, workers in analysis, the branch of mathematics beginning with calculus, were dealing with infinite trigonometric series: y ¼ ða1 cos x þ b1 sin xÞ þ ða2 cos 2x þ b2 sin 2xÞ þ ða3 cos 3x þ b3 sin 3xÞ þ . . . The “vibrating string controversy” engaging Leonhard Euler and others concerned how wide a class of functions can be represented in this form. The dispute exposed, beyond endemic deficiencies of rigor in the treatment of infinite series, lack of a common understanding about what is meant by a function. The ensuing nineteenth-century rigorization of analysis, besides banning any literal infinities or infinitesimals, explaining contexts containing the symbol ∞ without assuming it to denote anything in isolation, fixed on the maximally general notion of function, under which any correlation between inputs and outputs counts, as long as there is one and only one output per input. Improved rigor eventually led to consensus about the existence of trigonometric series representations.