From Unity of Truth: “Are all infinities the same?”
Well, it seems obvious that that there must be twice as many numbers if you count the half numbers as well as the whole numbers, and three times as many if you count the 1/3 numbers as well as the whole numbers – indeed six times as many if you count halves, thirds, and whole numbers.
But what is obvious is not always true.
There are no more half numbers than whole numbers, because if you line up all the half you can count them – using whole numbers — for as long as you want. So there aren’t any more.
Look, No Exclamation Points
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I Guess the New York Times Doesn’t Know About Singapore Math