GÖDEL ON CANTOR’S CONTINUUM PROBLEM II: CANTOR’S CONTINUUM PROBLEM

Abstract

Cantor’s continuum problem is the question: How many points are there on a line in Euclidean space? Cantor (Georg Cantor) believed that the number of points on a line, i.e. the cardinality of the continuum, is the fi rst infi nite cardinal that comes after the cardinality of the set of natural numbers. This is a rough formulation of Cantor’s continuum hypothesis. Cantor repeatedly tried to prove the continuum hypothesis during his career. However, his eff orts did not bear fruit. At the beginning of the 20th century, set theory was formulated as an axiomatic theory. That axiomatic theory is called Zermelo-Frenkel set theory with axiom of choice, i.e., ZFC set theory. Based on the results of Kurt Gödel and Paul Cohen, we know that the ZFC theory neither proves nor disproves Cantor’s hypothesis, leaving Cantor’s continuum problem unsolved. However, according to Gödel, Cantor’s continuum problem can be solved by extending ZFC set theory with new axioms that will give us a more complete insight into the structure of the universe of sets. This paper is the second in a series of two papers in which we aim to present some of Gödel’s proposals for an approach to solving Cantor’s continuum problem, as well as Gödel’s Platonist philosophical position underlying those proposals. In this paper, we aim to present Gödel’s proposals for new axioms and their consequences, together with the justifi cation that can be off ered in support of those principles. We will fi rst present Gödel’s reactions to the views according to which Cantor’s continuum problem is solved by proving the independence of CH. We will then consider Gödel’s opinion on the truth of CH. We will then state some of the strong axioms of infi nity and their consequences for Cantor’s problem. Finally, we will consider the axiom of constructibility as a candidate for a new axiom.