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: $\S 10.46 \ (3)$