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I was reading the book "Quantum Computing Since Democritus".

"The set of ordinal numbers has the important property of being well ordered,which means that every subset has a minimum element. This is unlike the integers or the positive real numbers, where any element has another that comes before it."

Unlike integers? Let's consider a set $\{1,2,3\}$ This has a minimum element.

What does the author want to say here?

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    $\begingroup$ I’m voting to close this question because it belongs to math.SE $\endgroup$
    – glS
    Commented Feb 7, 2021 at 14:45
  • $\begingroup$ Agreed; this question is reasonable, well-motivated, and well-formulated, but the connection to quantum computing is tenuous. It might be a good fit for math.stackexchange -link; it's recommended to ask the question there. $\endgroup$ Commented Feb 7, 2021 at 15:44
  • $\begingroup$ I agree that MSE is a better home for this question. BTW: I just noticed that the title asks a different question than the body of the post. $\endgroup$ Commented Feb 7, 2021 at 17:05

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An ordered set $X$ is well-ordered if every non-empty subset of $X$ has a least element.

As stated in the quote, the set $\mathbb{Z}$ of integers is not well-ordered (under the usual order relation). An example of a subset of the integers $\mathbb{Z}$ which fails to have a least element is

$$ \{\dots, -8, -6, -4, -2, 0\}, $$

another is $\mathbb{Z}$ itself.

The set $\{1, 2, 3\}$ is well-ordered since all seven non-empty subsets of it have a least element. Note that the quote does not deny well-ordering of $\{1, 2, 3\}$.

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