Definition:Order Type of Real Numbers
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Definition
Consider the ordered set $\struct {\R, \le}$ consisting of the set of real numbers under the usual ordering $\le$.
The order type of $\struct {\R, \le}$ is denoted $\lambda$ (lambda).
Historical Note
The use of $\lambda$ to denote the order type of $\struct {\R, \le}$ 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 $\struct {\Q, \le}$, has not been greatly studied in recent years.
$\lambda$ is in fact far more often used to denote an infinite cardinal.
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