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Fri, 29 May 2020
Infinite zeroes with one on the end
I recently complained about people who claim:
When I read something like this, the first thing that usually comes to mind is the ordinal number !!\omega+1!!, which is a canonical representative of this type of ordering. But I think in the future I'll bring up this much more elementary example: $$ S = \biggl\{-1, -\frac12, -\frac13, -\frac14, \ldots, 0\biggr\} $$ Even a middle school student can understand this, and I doubt they'd be able to argue seriously that it doesn't have an infinite sequence of elements that is followed by yet another final element. Then we could define the supposedly problematic !!0, 0, 0, \ldots, 1!! thingy as a map from !!S!!, just as an ordinary sequence is defined as a map from !!\Bbb N!!. [ Related: A familiar set with an unexpected order type. ] [Other articles in category /math] permanent link |