Definition:Well-Ordering
From ProofWiki
Definition
Let $\left({S, \preceq}\right)$ be an ordered set.
Then the ordering $\preceq$ is a well-ordering on $S$ iff $\preceq$ is well-founded.
If this is the case, then $\left({S, \preceq}\right)$ is referred to as a well-ordered set or woset.
Also see
- Well-Ordering is Total Ordering, which shows that every well-ordering is in fact a total ordering.
Sources
- W.E. Deskins: Abstract Algebra (1964): $\S 2.3$: Definition $2.3$
- Seth Warner: Modern Algebra (1965): $\S 14$
- Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.1$: Exercise $15$
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$
