Meaning of well-order | Babel Free

Definitions

A total order of some set such that every nonempty subset contains a least element.

Equivalents

Examples

“1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237, ̲X is well-order enriched iff every morphism set ̲X(X,Y) carries a well-order ≤_(XY) such that f≨_(XY)g⇒h•f≨_(XY)h•g for every h:Y→Z.”

“2001, Robert L. Vaught, Set Theory: An Introduction, Springer (Birkhäuser), 2nd Edition, Softcover, page 71, Some simple facts and terminology about well-orders were already given in and just before 1.8.4. Here are some more: In a well-order A, every element x is clearly of just one of these three kinds: x is the first element; x is a successor element - i.e., x has an immediate predecessor; or x is a limit element - i.e., x has a predecessor but no immediate predecessor. The structure (∅, ∅) is a well-order.”

“2014, Abhijit Dasgupta, Set Theory: With an Introduction to Real Point Sets, Springer (Birkhäuser), page 378, Definition 1226 (Von Neumann Well-Orders). A well-order X is said to be a von Neumann well-order if for every x∈X, we have x=y∈X|y<x (that is x is equal to the set Pred(x) consisting of its predecessors). Clearly the examples listed by von Neumann above, namely empty , empty, empty ,empty, empty ,empty,empty ,empty, … are all von Neumann well-orders if ordered by the membership relation "∈," and the process can be iterated through the transfinite. Our immediate goal is to show that these and only these are the von Neumann well-orders, with exactly one von Neumann well-order for each ordinal (order type of a well-order). This is called the existence and uniqueness result for the von Neumann well-orders.”

CEFR level