Of all the formal constructions of the set of the real numbers, Eudoxus reals are to me the most elegant, even if completely impractical.
The idea, due to Schnauel, is that any line in the plane, be its slope rational or irrational, can be described using only points with integer coordinates: taking the floor (or ceil or round or worse, doesn’t matter) of the y-values associated with integer x-values does characterize uniquely, asymptotically, any line. The slope can be recovered by linear regression (again, on an ideally infinite set of data).
The construction reverses this observation:
- An almost-homomorphism of the integers is defined to be a function whose deviance from being additive is bounded. Rounding a number never gives you an answer farther than 1 from the true value, so the rounding of a line is an example of almost-homomorphism, or, you could say, an almost-line.
- A real number is an almost-homomorphism modulo a bounded function, as an almost-line that never goes higher or lower than a certain height fits to the horizontal axis, that represents the number 0.
- It turns out this set admits operations that make it a complete ordered field. Proving this doesn’t take long but is very cumbersome.
A complete treatment is given in this article by Arthan, which notes that the important property of the integers is that they form an abelian group of which a subset is bounded if and only if is finite. This allows to generalize the construction to real matrices.
The construction isn’t due to Eudoxus, but his name was nonetheless given to it because of his theory of proportion, which inspired the author.