Why isn't $\{1,2,3\}$ well ordered? [closed]

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?

• I’m voting to close this question because it belongs to math.SE
– glS
Commented Feb 7, 2021 at 14:45
• 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. Commented Feb 7, 2021 at 15:44
• 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. Commented Feb 7, 2021 at 17:05

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 $$\{1, 2, 3\}$$.