Infinity

Infinity 500 BCE – 2030 CE

From Zeno's paradoxes to Cantor's transfinite ladder — how the endless was tamed into mathematics.

• Infinity as paradox450 BCE

  Zeno of Elea, Paradoxes. Zeno's paradoxes — Achilles and the tortoise, the dichotomy — argue that motion is impossible because any distance contains infinitely many sub-distances. Whether meant to defend Parmenides or merely to provoke, they force the first serious confrontation with the infinitely divisible.

• Potential, not actual, infinity350 BCE

  Aristotle, Physics. Aristotle resolves Zeno by distinguishing the potential infinite — a process that can always be continued — from the actual infinite, a completed endless whole, which he rejects as incoherent. This ban on completed infinities governs mathematics for two thousand years.

• Infinity as a tool for measure250 BCE

  Archimedes, The Method. Archimedes computes areas and volumes by exhausting curved figures with infinitely many slices, a method rediscovered only in 1906 in a palimpsest. He handles the infinite operationally while carefully proving results by the finite method of exhaustion — foreshadowing the calculus.

• The whole equals the part1638 CE

  Galileo, Two New Sciences. Galileo notices that the squares can be paired one-to-one with all the integers, so an infinite set can be matched with a proper part of itself. He concludes that 'equal', 'greater' and 'less' simply do not apply to infinities — the very paradox Cantor would later embrace.

• A symbol for the endless1656 CE

  John Wallis, Arithmetica Infinitorum. Wallis introduces the lemniscate ∞ as a sign for the infinite and treats infinitely small quantities freely to derive his famous product for π. The infinite acquires a notation and, with it, a working presence in algebra.

• Taming the infinitesimal1684 CE

  Newton & Leibniz, the calculus. Independently, Newton with his 'fluxions' and Leibniz with his differentials build a calculus that computes tangents and areas by reasoning about vanishingly small quantities. It is spectacularly powerful yet logically shaky — Bishop Berkeley soon mocks these infinitesimals as 'ghosts of departed quantities'.

• Limits replace infinitesimals1821 CE

  Cauchy, Cours d'analyse. Cauchy rebuilds the calculus on the rigorous notion of a limit, later sharpened into the epsilon–delta definition. The infinite is no longer a quantity but a controlled process of approach — restoring Aristotle's potential infinite with modern precision.

• Rehabilitating the actual infinite1851 CE

  Bolzano, Paradoxes of the Infinite. Published posthumously, Bolzano's essay defends the actual infinite as a legitimate object and studies infinite sets in their own right, noting the one-to-one pairings Galileo had found troubling. He prepares the ground on which Cantor will build.

• Infinities of different sizes1874 CE

  Georg Cantor, set theory. Cantor proves that the real numbers cannot be paired with the integers: some infinities are strictly larger than others. His diagonal argument and his transfinite cardinals turn the actual infinite into a rigorous hierarchy — a revolution Kronecker denounced but which reshaped mathematics.

• The continuum hypothesis1900 CE

  Hilbert, Paris problems. Hilbert opens his famous list with Cantor's continuum hypothesis — is there an infinity strictly between the integers and the reals? Defending set theory, he later declares that 'no one shall expel us from the paradise Cantor has created' for us.

• The infinite escapes the axioms1963 CE

  Paul Cohen, forcing. Building on Gödel, Cohen invents forcing to prove the continuum hypothesis independent of the standard axioms of set theory: it can be neither proved nor disproved. The size of the continuum turns out to be a matter of choice, not fact — and Cohen wins the Fields Medal.