Well-Ordering of Class of All Ordinals under Subset Relation
Jump to navigation
Jump to search
Theorem
Let $\On$ denote the class of all ordinals.
$\On$ is well-ordered by the subset relation such that the following $3$ conditions hold:
\((1)\) | $:$ | 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. |
Proof
We have that Class of All Ordinals is $g$-Tower.
By Zero is Smallest Ordinal, $0$ is the smallest element of $\On$.
We identify the natural number $0$ via the von Neumann construction of the natural numbers as:
- $0 := \O$
The result then follows directly from $g$-Tower is Well-Ordered under Subset Relation.
$\blacksquare$
Sources
- 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 $1.11$