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