Overview
Definition
The following defintion follows that found in Krantz.
A cut, {% \mathcal{C} %} is a nonempty subset of the rational numbers {% \mathbb{Q} %} such that the following holds
- if {% s \in \mathcal{C} %} and {% t < s %} then {% t \in \mathcal{C} %}
- if {% s \in \mathcal{C} %} then there an element of {% \mathcal{C} %}, {% u %} such that {% u > s %}
- There is an {% x \in \mathbb{Q} %} such that for {% c \in \mathcal{C} %}, {% c < x %}
The set of cuts is defined to be the set of real numbers.
Examples
- The rational number {% q %} is represented by the set of rationals {% p %} such that {% p < q %}.
- The irrational number {% \sqrt{2} %} is represented by the set of rationals {% p %} such that {% p < \sqrt{2} %}.