Definition:Well-Ordered Set
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Definition
Let $\left({S, \preceq}\right)$ be an ordered set.
Then $\left({S, \preceq}\right)$ is a well-ordered set (or woset) if the ordering $\preceq$ is well-founded.
That is, if every $T \subseteq S: T \ne \varnothing$ has a minimal or first element.
That is, $\exists a \in T: \forall x \in T: a \preceq x$.
Note the every in the above.
Also see
- Well-Ordering is Total Ordering, which shows that every woset is in fact a toset.
- Results about well-orderings and well-ordered sets can be found here.
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 17$: Well Ordering
- Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents
- A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.5$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$: Exercise $4$, $\S 8$
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$
