Well-Ordering of Class of All Ordinals under Subset Relation
Jump to navigation Jump to search
Let $\On$ denote the class of all ordinals.
|\((1)\)||$:$||the smallest ordinal is $0$|
|\((2)\)||$:$||for $\alpha \in \On$, the immediate successor of $\alpha$ is its successor set $\alpha^+$|
|\((3)\)||$:$||every limit ordinal is the union of the set of smaller ordinals.|
We have that Class of All Ordinals is $g$-Tower.
- $0 := \O$
The result then follows directly from $g$-Tower is Well-Ordered under Subset Relation.
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.11$