What does it mean for a set to be well ordered or ordered? Is there a connection between a well-ordered set and an ordered set ? If there is, then can one be derived from the other? Or if there is no connection then what is the difference between them.
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Are the integers ordered? well-ordered? Are the naturals ordered? well-ordered? – Eric Towers Jun 17 '19 at 08:43
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Well ordered is a subset of ordered which also includes partially ordered, cyclic orders etc etc. – Phil H Jun 17 '19 at 09:07
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Have tried looking at the definitions in the Wikipedia? – jjagmath Mar 14 '24 at 15:03
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Assuming you already understand what an order relationship means:
Let $S$ be an ordered set. We say that $S$ is well-ordered if, and only if, $\forall A \subseteq S, A \neq \emptyset, \exists x\in A$ such that $x=min(A)$
If you try to check by yourself whether $\mathbb{R}$, $\mathbb{Z}$ and $\mathbb{N}$ are well ordered sets, and get it right, you can say you understood the idea. It's pretty basic
David
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So in that sense, the real numbers are ordered but not well ordered. On the other hand, the positive integers are both ordered and well ordered. – Ayan Shah Jun 17 '19 at 11:05
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You can define order relationships in both $\mathbb{C}$ and $\mathbb{R}^2$ (or any other set you want) When we say $\mathbb{C}$ is not ordered, we actually mean that no order relationship defined on $\mathbb{C}$ will have the same algebraic properties as the usual order relation in $\mathbb{R}$ Here's a proof https://math.stackexchange.com/questions/312204/prove-that-field-of-complex-numbers-cannot-be-equipped-with-an-order-relation Anyway, I don't see why that is different than $\mathbb{R}^2$ – David Jun 17 '19 at 11:16
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