Surjection from Aleph to Ordinal

From ProofWiki
Jump to navigation Jump to search

This article is not under active maintenance.

While the contents of this page could be useful, they are currently not being maintained.

The correctness, lay-out and usefulness of the article may be compromised, so use whatever you get from here with caution.


Let $x$ and $y$ be ordinals.

Suppose that:

$0 < y < \aleph_{x+1}$

Then there is a surjection:

$f : \aleph_x \to y$


$y < \aleph_{x+1}$, 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.