 An example: when we have defined the rationals, and then defined the reals as the collection of Dedekind cuts, how do we define the square root of 2? It is reasonably straightforward to show that \((\{x \in \mathbf{Q} \mid x \lt 0 \vee x^2 \lt 2\}, \{x \in \mathbf{Q} \mid x \gt 0 \amp x^2 \ge 2\})\) is a cut and (once we define arithmetic operations) that it is the positive square root of two. When we formulate this definition, we appear to presuppose that any property of rational numbers determines a set containing just those rational numbers that have that property.