Def/Well ordered

Logical Connections

This definition logically relies on the following definitions and statements: Def/Binary relation, Def/Total order, Def/Minimal element

The following statements and definitions logically rely on the material of this page: Def/Ordinal number

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Suppose that $(X, \leq)$ is a well-ordered set. Then, $X$ itself must have a minimal element; we may call it $x_{min}$.

Furthermore, suppose that $x \in X$. Let $S = \{y \in X \mbox{ such that } x < y \}$. Then, $S$ has a minimal element. In other words, there is a "smallest element bigger than $x$"; since $(X, \leq)$ is totally ordered, it follows that $S$ has a unique minimal element. Therefore, we may safely call this the "successor" of $x$ in $S$. Every element of a well-ordered set has a successor.

This article has been written by: User:Marty