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Let $(A,<)$ be well-ordered set, using <, how can one define well-order on set of finite sequences?

(I thought using lexicographic order)

Thank you!

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What's the problem with the lexicographic order? –  Isaac Solomon Jan 5 '13 at 0:32
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Haim told me to say hi. :-) –  Asaf Karagila Jan 5 '13 at 0:34
    
@AsafKaragila, now his question are famous! :) –  17SI.34SA Jan 5 '13 at 0:45
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I would have written better questions! :-) –  Asaf Karagila Jan 5 '13 at 0:46
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1 Answer

up vote 3 down vote accepted

The lexicographic order is fine, but you need to make a point where one sequence extends another -- there the definition of the lexicographic order may break down. In this case you may want to require that the shorter sequence comes before the longer sequence.

Generally speaking, if $\alpha$ and $\beta$ are two well-ordered sets, then we know how to well-order $\alpha\times\beta$ (lexicographically, or otherwise). We can define by recursion a well-order of $\alpha^n$ for all $n$, and then define the well-order on $\alpha^{<\omega}$ as comparison of length, and if the two sequences have the same length, $n$, then we can compare them by the well-order of $\alpha^n$.

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