Three forms of mathematical induction: Difference between revisions

Revision as of 00:09, 7 March 2006

Revision as of 00:11, 7 March 2006

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* The substantial part of the proof is the ''second'' case. The second case is just the statement of the ordinary triangle inequality, and is different in different metric spaces. It is part of the proof that a particular function ''d'' is in fact a metric. In some cases is difficult or otherwise onerous. Usually the other aspects of proving that ''d'' is a metric are trivial (i.e. that ''d''(''x'', ''x'') = 0 for all ''x'', and that ''d'' is symmetric).

====PolyaPólya's proof that there is no "horse of a different color"====

In the middle of the 20th century, a commonplace colloquial locution to express the idea that something is unexpectedly different from the usual was "''That's'' a horse of a different color!". [[George PolyaPólya]] posed the following exercise: find the error in the following argument, which purports to prove by mathematical induction that all horses are of the same color:

:If there's only ''one'' horse, there's only one color.