Product of Finite Sets is Finite/Proof 2
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Theorem
Let $S$ and $T$ be finite sets.
Then $S \times T$ is a finite set.
Proof
Let $\card S$ denote the cardinal number of $S$.
Let $\cdot$ denote ordinal multiplication.
By Cardinal Product Equinumerous to Ordinal Product, it follows that $S \times T \sim \card S \cdot \card T$.
But then $\card S$ and $\card T$ are members of the minimally inductive set.
Therefore, $\card S \cdot \card T \in \omega$ by Natural Number Multiplication is Closed.
Since $S \times T$ is equinumerous to a member of the minimally inductive set, it follows that $S \times T$ is finite.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.29 \ (2)$