Cardinal of Finite Ordinal
Jump to navigation
Jump to search
This article needs to be linked to other articles.
In particular: Predecessor of Finite Ordinal is Finite Ordinal/Element of Finite Ordinal is Finite Ordinal.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
Theorem
Let $n$ be a finite ordinal.
Let $\left|{n}\right|$ denote the cardinal number of $n$.
Then:
- $\left|{n}\right| = n$
Proof
Since $n$ is an ordinal, it follows that $\left|{n}\right| \le n$ by Cardinal Number Less than Ordinal: Corollary.
In particular: Predecessor of Finite Ordinal is Finite Ordinal/Element of Finite Ordinal is Finite Ordinal.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this in more detail, feel free to use the talk page.
Once you have done so, you can remove this instance of {{MissingLinks}} from the code.
Hence, $\left|{n}\right|$ is also a finite ordinal.
Since $n$ is an ordinal, it also follows that $n \sim \left|{n}\right|$ by Ordinal Number Equivalent to Cardinal Number.
 |
A specific link is needed here: Equality of Finite Ordinals? (discuss) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by searching for it, and adding it here. |
By Equality of Natural Numbers, it follows that $n = \left|{n}\right|$.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.19$