The Continuum Hypothesis:

A Look at the 20th Century History of the Real Numbers

from Cantor to Cohen/Scott/Solovay.

Initial Questions (Optional PreTest) [3:50-4:00]

Pre XXth Century Western Views of the Real Number / Continuum- Condensed  Soup  Version:
(Thanks to The MacTutor History of Mathematics Archive)

I. "Greek" Views: [4:05, 4:10]
Greek Numbers and Fractions:
Measurement and Finding Common Units
Irrationality: No common unit. The square root of 2.
Eudoxus (408-355 BCE) / Euclid (325-265 BCE)
Theory of Proportion.
Archimedes(287-212 BCE)
Estimation of Pi as a ratio:
Circumference to Diameter of a Circle.

II. Pre-Calculus and Early Calculus Views: [4:10, 4:15]
Stevin: 1548-1620
Napier: 1550-1617
Briggs: 1561-1630
Descartes: 1596-1650
Newton 1643-1727/ Mercator 1620-1687:
Computation of hyperbolic logarithms using decimals.
Newton & Leibniz 1646-1716 "limits and functions"

III. The Age of Development and Conventions [4:15,4:20]
Bolzano 1781- 1848 (1817 paper)
Dirichlet: 1805-1859

IV. The Age of Critical Awareness and Foundations [4:20, 4:25]
Weierstrass: 1815- 1897
Dedekind: 1831-1916
Frege: 1848-1925
Peano: 1858-1932

V. Cantor 1845-1918: Investigation of discontinuities with Fourier series and Set Theory Beginnings. [4:25,4:35]
  • Any infinite subset of the natural numbers or the integers is countable.
  • The rational numbers are a countable set.
  • "Godel counting" argument.
  • The algebraic numbers are countable.

  • [ Another first type of diagonal argument.] 1874
  • Cantor's proof that the number of points on a line segment are uncountable. (1874)
  • A decimal based proof that there is an uncountable set of real numbers.(similar to 1891 proof)
  • The set of {0,1} valued  sequences in "uncountable."
  • There is no onto function from R, the set of real numbers, to P(R), the set of all subsets of the real number.
  • There are sets which are larger than the reals.
  • The rational numbers between 0 and 1 have  "measure" zero.
  • Any countable set of real numbers has "measure" zero.

  • VI.  The XXth Century: An Age of Exploration and Discovery. [4:35- 4:50]
    Hilbert: (1862-1943) (Finitistic Formalization of Arithmetic)
    The continuum hypothesis problem was the first of Hilbert's famous 23 problems delivered to the Second International Congress of Mathematicians in Paris in 1900. Hilbert's famous speech The Problems of Mathematics challenged (and still today challenge) mathematicians to solve these fundamental questions

    Brouwer: (1881-1966) (Rejection of the law of excluded middle for infinite sets) He rejected in mathematical proofs the Principle of the Excluded Middle, which states that any mathematical statement is either true or false. In 1918 he published a set theory, in 1919 a measure theory and in 1923 a theory of functions all developed without using the Principle of the Excluded Middle.

    Godel: (1906-1978)
    Godel Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (1940)
    Gödel showed, in 1940, that the Axiom of Choice and/ or the Continuum Hypothesis cannot be disproved using the other axioms of set theory

    Cohen: (1934- )
    It was not until 1963 that Paul Cohen proved that the Axiom of Choice is independent of the other axioms of set theory. Cohen used a technique called "forcing" to prove the independence in set theory of the axiom of choice and of the generalised continuum hypothesis.

     Dana Scott/ Solovay:
    Models for the real numbers based on Probability-Measure Theory.


    Philosophical Introduction to Set Theory by Stephen Pollard
    The Mathematical Experience by Philip J. Davis and Reuben Hersh
    What is mathematical logic? by J.N. Crossley et al.
    Set  Theory and the Continuum Hypothesis by Raymond M. Smullyan and Melvin Fitting
    Intermediate Set Theory by F.R. Drake and D. Singh