This article presents a fascinating thought experiment: imagine breaking a ruler at the 4 cm mark, leaving the left piece with a jagged edge. What is the exact value of that point? This simple question leads into a deep discussion of Dedekind cuts, the construction of real numbers, and the paradoxes that arise when abstract mathematics meets physical intuition. The author explores how the real number line, as defined by Dedekind cuts, does not perfectly map onto physical measurements, revealing a gap between mathematical theory and empirical reality. This piece is a must-read for mathematicians, philosophers, and computer scientists interested in the foundations of mathematics and the nature of the continuum. It challenges readers to think critically about the assumptions underlying our mathematical models of the world.
A thought experiment on Dedekind cuts and the continuum, using a broken ruler to question the nature of real numbers.