I don't get it, a book i read on set theory states that every set can be well-ordered, while another book i'm reading, about proofs, says that for example the open interval $(0,1)$ is not well-ordered. How is this possible? I know that the definition states that a non-empty set $S$ is well-ordered if every non-empty subset of $S$ has a least element, but, how can they contradict each other?
2026-04-18 18:37:13.1776537433
Every set can be well-ordered or not?
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You may confuse the well-orderability and being well-ordered.
The axiom of choice proves every set is well-orderable, that is, we can impose a well-order over any set. On the other hand, $(0,1)$ with the natural order is not well-ordered. You may well-order the set $(0,1)$, but the well-order does not respect the natural order of the real numbers.