Class of All Cardinals Contains Minimally Inductive Set
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Theorem
Let $\NN$ denote the class of all cardinal numbers.
Then:
- $\omega \subseteq \NN$
where $\omega$ denotes the minimally inductive set.
Proof
Suppose $n \in \omega$.
By Cardinal of Finite Ordinal, $n = \card n$.
By Cardinal of Cardinal Equal to Cardinal/Corollary, $n \in \NN$.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.39$