Existence of Disjoint Well-Ordered Sets Isomorphic to Ordinals

Theorem

Let $\alpha$ and $\beta$ be ordinals.

Then there exist two well-ordered sets $a$ and $b$ such that:

$a$ and $b$ are order isomorphic to $\alpha$ and $\beta$ respectively

$a$ and $b$ are disjoint.

Proof

Let $a$ and $b$ be defined as:

\(\ds a\)

\(:=\)

\(\ds \alpha \times \set 0\)

\(\ds b\)

\(:=\)

\(\ds \beta \times \set 1\)

Then let $a$ and $b$ be ordered by their first coordinate.

The result follows.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 6$ Ordinal arithmetic