Lesson 3

The hypothesis stating that there are no sets of cardinality greater than countable and less than the continuum is known as the Continuum Hypothesis and as the first problem of Hilbert: formulated by Georg Cantor in 1877, in 1900 David Hilbert made it the first in his list of twenty-three open problems. For many decades, this problem was considered one of the most difficult open problems in mathematics. Formally, it is written as: \[\aleph_{1}=2^{\aleph_0}\ .\] Here \(\aleph_{0}\) is the first infinite cardinal (the minimum cardinality of an infinite set, i.e., the cardinality of a countable set), \(\aleph_{1}\) is the next one.

Gödel showed that the negation of the continuum hypothesis is unprovable in ZFC (the Zermelo–Fraenkel axiom system), and Cohen proved that the hypothesis itself is unprovable: there exist models of set theory in which the hypothesis is true, and others where it is false.