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Why isn't the ordinary string comparison a valid well ordering of the real numbers? That is, write the number as a string (possibly infinite in size):

a = "1.414"
b = "1.41421..." sqrt(2)

Then use some string comparison. If the numbers are distinct they should have a distinct decimal representation at some finite position.

Asaf Karagila
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F. Bruno Dias
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    It may be a total ordering (be careful about things like $1 = 0.\bar{9}$), but it doesn't look like a well-ordering to me. – Tzimmo Apr 02 '25 at 16:34
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    Well ordering would require that subsets, such as the non-negative reals, have a least element. – lulu Apr 02 '25 at 16:34
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    {1.1, 1.01, 1.001, 1.0001, ...} doesn't have a least element. – JonathanZ Apr 02 '25 at 16:47
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    You said to "use some string comparison" --- OK, I'll use the lexicographic ordering. Then $0.1>0.01>0.001>\dots$. So the result is not a well-ordering. – Andreas Blass Apr 02 '25 at 16:48
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    Thank you lulu, JonathanZ and Andreas, I missed the requirement of having a least element! – F. Bruno Dias Apr 02 '25 at 16:54

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