Surjection from Aleph to Ordinal
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Theorem
Let $x$ and $y$ be ordinals.
Suppose that:
- $0 < y < \aleph_{x+1}$
Then there is a surjection:
- $f : \aleph_x \to y$
Proof
- $y < \aleph_{x+1}$, then $y < \aleph_x \lor y \sim \aleph_x$ by Ordinal Less than Successor Aleph.
In either case, $\left|{ y }\right| \le \aleph_x$ by Ordinal in Aleph iff Cardinal in Aleph and Equivalent Sets have Equal Cardinal Numbers.
The existence of the surjection follows from Surjection iff Cardinal Inequality.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.46 \ (3)$