= Dedekind cut
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{title2=$A\subsetneq\mathbb Q$}
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A <Dedekind cut> can be represented by a nonempty proper <subset> $A$ of the <rational numbers> satisfying two conditions: if $q\in A$ and $r<q$ is rational, then $r\in A$; and $A$ has no <maximum>. The complementary <set> contains all the rational values on the other side of the boundary. Using the convention that the lower <set> never includes its boundary avoids two representations of a rational boundary.
The <set> $\{q\in\mathbb Q:q<0\}$ represents zero. The <set> $\{q\in\mathbb Q:q<0\text{ or }q^2<2\}$ represents $\sqrt2$, even though no <rational number> occupies that boundary by the <irrationality of the square root of two>. Thus a <Dedekind cut> describes an <irrational number> entirely using rational comparisons. Ordering these cuts by the <subset> relation and defining their arithmetic gives a construction of the <real numbers>. https://www.gutenberg.org/files/21016/21016-pdf.pdf[Dedekind, Continuity and Irrational Numbers, §§IV–VI: cuts, continuity, and arithmetic]{ref}.
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