Well-Ordering on Set is Proper Well-Ordering
Jump to navigation
Jump to search
Theorem
Let $\struct {S, \preccurlyeq}$ be a well-ordered set.
Then $\preccurlyeq$ is a proper well-ordering.
Proof
By definition, a proper well-ordering is a well-ordering on a class such that:
- every proper lower section of $S$ is a set.
We have a fortiori that a proper lower section of $S$ is a subclass of $S$.
But here we have that $S$ is a set.
The result follows from Subclass of Set is Set.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries