How do you build an irrational number out of nothing but fractions? Dedekind’s answer: cut the rationals into two sets — everything too small, and everything too big. The gap between them is the number. Slide to sharpen the cut and watch it close in on √2, a number no fraction can name.
The cut is the pair of sets L={q : q<0 or q²<2} and U={q : q>0 and q²>2}. Neither has a rational at the boundary — that missing point is √2, and its existence as a real number just IS the cut. The instrument narrows L and U by bisection; a fail-loud self-check throws unless after 60 refinements the boundary squared is within 1e−12 of 2 and within 1e−9 of √2, with L’s top always below U’s bottom (the cut never crosses).
Bisection is a way to WATCH the cut close; Dedekind’s construction (1872) doesn’t compute — it DEFINES the real as the set itself. Floating point is a finite stand-in for the infinite set of rationals; the honesty is that the boundary is never a fraction.