Ordinal Number Equivalent to Cardinal Number
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Theorem
Let $x$ be an ordinal.
Let $\card x$ denote the cardinal number of $x$.
Then:
- $x \sim \card x$
where $\sim$ denotes set equivalence.
Proof
From Set is Equivalent to Itself:
- $x \sim x$
Therefore, $x$ is equivalent to some ordinal.
By Condition for Set Equivalent to Cardinal Number:
- $x \sim \card x$
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.11$