Constructing numbers

A miracle of mathematics lies in the fact that multiple, vastly different constructions of the real numbers all eventually agree on the same result. While the destination is identical, the paths taken by Dedekind cuts, Cauchy sequences, and Surreal numbers reveal deeply contrasting philosophies about the nature of a number itself, encoding unique intuitions about order, approximation, and creation. 

Dedekind cuts begin with the observation that while the rational numbers are ordered, they remain incomplete. This method defines a number by splitting the rationals into two distinct sets containing everything less than and everything greater than the value in question. The number itself serves as the static boundary between these sets. In this framework, nothing moves or converges, and nothing is approximated over time. Instead, a real number is a separator, making this approach fundamentally order-theoretic. It answers the question of where gaps must be filled given an existing ordering, treating irrational numbers as unavoidable consequences of that order rather than limits of a process. This conceptual clarity makes completeness feel almost tautological by emphasizing structure over computation, though the method remains awkward for actual calculation and is difficult to generalize beyond ordered fields.

In contrast, Cauchy sequences approach the problem from a procedural perspective. Rather than cutting the rationals in half, this method asks when a sequence of rationals behaves as if it is converging, even if the destination does not exist within the rational system. A real number is thus defined as an equivalence class of Cauchy sequences whose terms grow arbitrarily close to one another. Here, numbers emerge from approximation over time, where a value like the square root of two exists because a sequence of rationals behaves as though it is closing in on a target. This analytic construction aligns naturally with limits and continuity, extending smoothly to other metric spaces. However, it relies heavily on infinite processes and requires the conceptual weight of equivalence classes, which can make the concept of completeness feel less transparent than the structural approach of a cut.

Surreal numbers represent a third philosophy that departs from the goal of merely completing the rationals. They start with nothing and define each number recursively in terms of the elements that come before and after it. While this looks superficially like a Dedekind cut, the underlying logic is different because it grows the structure itself rather than partitioning a fixed one. This process yields not only the real numbers but also ordinals, infinitesimals, and infinite numbers, making the reals feel like a small and tame subset of a much larger universe. This approach is astonishingly general and unifies magnitude with infinity, though it is often considered overkill for classical analysis and remains technically demanding to connect to standard computation.

Ultimately, these three constructions embody fundamentally different views of mathematical reality. Dedekind cuts suggest that if a gap can be described, it must exist as a boundary. Cauchy sequences propose that if a process stabilizes, it must converge to a limit. Surreal numbers argue that if something can be consistently placed between other objects, it exists as a comparison. These methods shape the perception of the real line, making it feel rigid and inevitable through cuts, dynamic and analytic through sequences, or provisional and creative through the surreals. While Cauchy sequences are practical for analysis and Dedekind cuts are clarifying for foundations, surreal numbers expand the very definition of what a number can be. They do not compete so much as they illuminate different constraints hiding inside the same mathematical object.