Product of Finite Sets is Finite/Proof 2

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: $\S 10.29 \ (2)$