Ordinals have No Zero Divisors

Theorem

Let $x$ and $y$ be ordinals.

Then:

$\paren {x \cdot y} = 0 \iff \paren {x = 0 \lor y = 0}$

Proof

Necessary Condition

Suppose that $\paren {x \cdot y} = 0$ and that $x \ne 0$.

By Ordinal Multiplication by Zero:

$\paren {x \cdot 0} = 0$

Therefore:

$\paren {x \cdot y} = \paren {x \cdot 0}$

Thus, by Ordinal Multiplication is Left Cancellable, we have that $y = 0$.

$\Box$

Sufficient Condition

If $x = 0$, then by Leibniz's Rule:

$\paren {x \cdot y} = \paren {0 \cdot y}$

If $y = 0$, then also by Leibniz's Rule:

$\paren {x \cdot y} = \paren {x \cdot 0}$

By Ordinal Multiplication by Zero, both $\paren {0 \cdot y} = 0$ and $\paren {x \cdot 0} = 0$.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.22$