What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

§4. THE CONTINUUM HYPOTHESIS

The Hypothesis of the Continuum, now usually known as the Continuum Hypothesis, states that the cardinal of the set of all real numbers is the smallest infinite cardinal greater than that of the integers. It is now known that the Continuum Hypothesis is neither true nor false, butundecidable. In order to understand what this means, we must briefly recall the axiomatic method (p. 214). The axiomatic method specifies a mathematical object by stating an explicit system of conditions, axioms, that the object is required to satisfy. This focuses attention on the abstract relationships between that object and others, rather than on the raw materials from which it is “built.” Simple presentations of set theory assume that notions such as “set” are defined, and described how to manipulate them. In order to set up a rigorous framework in which to discuss the Continuum Hypothesis, it is necessary to specify a system of axioms for set theory.

In 1964 Paul Cohen proved that the truth of the Continuum Hypothesis depends upon which axioms for set theory are chosen. The situation is similar to that for geometry. The truth or falsity of Euclid’s parallel axiom depends upon the type of geometry: there is a “Euclidean” geometry for which it is true, but there are also “non-Euclidean” geometries for which it is false (see p. 218). Similarly, there are “Cantorian” set theories in which the Continuum Hypothesis is true and “non-Cantorian” ones in which it is false. Earlier Kurt Godel had proved that the Continuum Hypothesis is true in some axiomatizations of set theory. Using a new technique called “forcing,” Cohen proved that in other axiomatizations it is false. In particular, there is no distinguished choice of axioms that leads to a unique ’natural’ theory of sets.