An Analysis of the Continuum Hypothesis

This paper analyzes the Continuum Hypothesis, that the cardinality of a set of real numbers is either finite, countably infinite, or the same as the cardinality of the set of all real numbers. It argues (i) that the real numbers are as similar to the natural numbers as possible in the sense that the relationship between any general method of deciding membership of a set of real numbers and the cardinality of the set should be a natural generalization of the case of the same relationship in the case of a set of natural numbers; and (ii) that CH is a very strong choice principle that is maximally efficient as a principle for deciding whether a real number is in a set of real numbers in the sense that it is uniform in deciding membership for every real number in a countable number of steps. The approach taken is to formulate principles equivalent to or weaker than the Continuum Hypothesis and to use techniques from computer science (infinite binary search), information theory, and set theory to prove theorems that support theses (i) and (ii).