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.] 1874Cantor'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.
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.
Web Resources:
Infinite Ink:
The Continuum Hypothesis by Nancy McGough
AkiHiro
KanaMori, The Mathematical Development of Set Theory from Cantor to Cohen,
The Bulletin of Symbolic Logic, March 1996.
The
Continuum Hypothesis (I) MSRI Berkeley Lectures (2000) by W. Hugh Woodin
The
Continuum Hypothesis (II) MSRI Berkeley Lectures (2000) by W. Hugh Woodin
The
Continuum Hypothesis (I) AMS Notices Paper (2001) W. Hugh Woodin
The
Continuum Hypothesis (II) AMS Notices Paper (2001) W. Hugh Woodin
Lectures
at MSRI on Current work related to the Continuum Hypothesis
May 29 - June 1, 2001
Reading:
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