Pre XXth Century Western Views of the Real Number / Continuum-
Condensed Soup Version:
(Thanks to The MacTutor History of Mathematics Archive)
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.
Estimation of Pi as a ratio:
Circumference to Diameter of a Circle.
Newton 1643-1727/ Mercator 1620-1687:
Computation of hyperbolic logarithms using decimals.
Newton & Leibniz 1646-1716 "limits and functions"
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 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.
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