Researched
Real Analysis (Rigorous Calculus)
Bolzano, Cauchy and Weierstrass (1817-1872) replaced vague 'infinitely small' quantities with exact limits and built the real numbers from fractions, so calculus rested on firm ground.
Open in the interactive tree →Calculus had worked for more than a century on the loose idea of infinitely small quantities. Bolzano (1817) and Cauchy (Cours d'analyse, 1821) defined limits and continuity more carefully, Karl Weierstrass gave the exact definition of a limit that is still taught today (1861), and between 1869 and 1872 Méray, Heine, Cantor and Dedekind constructed the real numbers themselves. This exact footing led to Cantor's set theory (1874) and to Lebesgue's integral, the base of modern probability.
Prerequisites
- Calculus (Newton & Leibniz)1684Rigorous analysis puts the limits and numbers behind calculus on exact definitions.
Unlocks
- Set Theory & Formal Logic1874Cantor's sets of real numbers rest on the exact construction of the reals (1869-72).
- Measure-Theoretic Probability1902-1933Lebesgue's integral extends the exact Riemann integral of rigorous analysis.