Continuum Problem

the problem that consists in proving or disproving by the methods of set theory the following statement, called the continuum hypothesis: the cardinal number of the continuum is the first cardinal that exceeds the cardinal number of the set of natural numbers. The generalized continuum hypothesis states that for any set P the first cardinal that exceeds the cardinal number of P is the cardinal number of the set of all subsets of P.

The continuum hypothesis was stated by G. Cantor in the early 1880’s. Cantor himself and many other outstanding mathematicians of the turn of the 20th century attempted, without success, to prove the continuum hypothesis. This lack of success convinced a number of eminent mathematicians (the French mathematicians R. Baire and H. Lebesgue, the Soviet mathematician N. N. Luzin) that the continuum problem could not be solved by traditional set-theoretic methods. This conviction was decisively confirmed by the precise methods of mathematical logic and axiomatic set theory. In 1936, K. Gödel proved that the generalized continuum hypothesis is consistent with a certain natural axiomatization of set theory and consequently cannot be disproved by traditonal methods. Finally, in 1963 the American mathematician P. Cohen, using what he called the forcing method, succeeded in proving that the negation of the continuum hypothesis is also consistent with this axiomatization, so that the continuum hypothesis cannot be proved using ordinary set-theoretic methods. Followers of Cohen subsequently obtained numerous results by the forcing method that have shed light on the role of the continuum hypothesis and the generalized continuum hypothesis, as well as on the relation of the hypotheses to other set-theoretic principles.

The results obtained indicate that at the present stage of the development of set theory it is possible to approach the foundations of this science in different ways that yield essentially different answers to such natural problems of set theory as the continuum problem.