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**1 - 4**of**4**### THE INSTRUCTOR’S GUIDE TO REAL INDUCTION

"... 1.1. “Induction is fundamentally discrete... ” 1 1.2....is dead wrong! 2 ..."

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### REAL INDUCTION

"... An unpleasant feature of modern life is that people often tell you certain things are impossible that you wonder, or even strongly suspect, should indeed be possible. I want to talk (of course) about a mathematical instance of this idea. Ironically, mathematics is one of the very few fields in which ..."

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An unpleasant feature of modern life is that people often tell you certain things are impossible that you wonder, or even strongly suspect, should indeed be possible. I want to talk (of course) about a mathematical instance of this idea. Ironically, mathematics is one of the very few fields in which it is possible to prove – really prove – that something is impossible. However, in my experience impossibility proofs in mathematics often depend sensitively on the precise set of hypotheses. For instance, most students of mathematics know that it is not possible to trisect a 60 degree angle. What this means of course is that it is not possible using a compass and a straightedge. However, if you are allowed to take your straightedge and mark two points on it, then it becomes possible to trisect a 60 degree angle, and in fact any angle. (Keyword: neusis constructions.) The moral here is that, when you are told something is impossible in mathematics, you are actually being told that it is impossible under certain precise conditions. This does not necessarily mean you should give up. Often there are other reasonable conditions under which the impossible becomes possible. Sometimes these