Equality of Successors implies Equality of Ordinals
Jump to navigation Jump to search
Let $\On$ denote the class of all ordinals.
- $\forall \alpha, \beta \in \On: \alpha^+ = \beta^+ \implies \alpha = \beta$
From Class of All Ordinals is Well-Ordered by Subset Relation:
- $\alpha^+$ is the immediate successor of $\alpha$
- $\beta^+$ is the immediate successor of $\beta$
and no two distinct elements of $\On$ can have the same immediate successor.
The result follows.
- 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 $1.17$