Definition:Order Type of Rational Numbers

Definition

Consider the ordered set $\struct {\Q, \le}$ consisting of the set of rational numbers under the usual ordering $\le$.

The order type of $\struct {\Q, \le}$ is denoted $\eta$ (eta).

Historical Note

The use of $\eta$ to denote the order type of $\struct {\Q, \le}$ dates back to the writings of Georg Cantor and Felix Hausdorff.

According to Winfried Just and Martin Weese in their Discovering Modern Set Theory. I: The Basics of $1996$, this order type, along with that for $\struct {\R, \le}$, has not been greatly studied in recent years.

$\eta$ is in fact far more often used to denote a general ordinal.

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations