# 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 $\left|{ S }\right|$ denote the cardinal number of $S$.

Let $\cdot$ denote ordinal multiplication.

By Cardinal Product Equinumerous to Ordinal Product, it follows that $S \times T \sim \left|{ S }\right| \cdot \left|{ T }\right|$.

But then $\left|{ S }\right|$ and $\left|{ T }\right|$ are members of the minimal infinite successor set.

Therefore, $\left|{ S }\right| \cdot \left|{ T }\right| \in \omega$ by Natural Number Multiplication is Closed.

Since $S \times T$ is equinumerous to a member of the minimal infinite successor set, it follows that $S \times T$ is finite.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.29 \ (2)$