- #26

mathwonk

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*last*I can be at peace with these previously fuzzy statements! I still wonder how anyone can think they know anything about calculus who cannot define, or at least characterize by axioms, the real numbers. If you don't know what a real number is, why bother to state theorems about them, e.g. what do you mean when you invoke the intermediate value theorem? Of course the student is not at fault, since they have been taught by a teacher who did not bother to explain what was going on. The rubber hits the road with a student who thinks for instance that all real numbers are integers, or at best that they are all finite decimals that can be displayed on their calculator. With such an understanding, of course all the theorems of calculus are actually false!