Definition:Well-Ordering/Definition 1
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Definition
Let $\struct {S, \preceq}$ be an ordered set.
The ordering $\preceq$ is a well-ordering on $S$ if and only if every non-empty subset of $S$ has a smallest element under $\preceq$:
- $\forall T \subseteq S, T \ne \O: \exists a \in T: \forall x \in T: a \preceq x$
Also see
- Results about well-orderings can be found here.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 17$: Well Ordering
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction: Exercise $15$
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Order