Continuum hypothesis

The ‘continuum hypothesis’ (CH) asserts that there is no set intermediate in cardinality (‘size’) between the set of real numbers (the ‘continuum’) and the set of natural numbers. Since the continuum can be shown to have the same cardinality as the power set (that is, the set of subsets) of the natural numbers, CH is a special case of the ‘generalized continuum hypothesis’ (GCH), which says that for any infinite set, there is no set intermediate in cardinality between it and its power set.

Cantor first proposed CH believing it to be true, but, despite persistent efforts, failed to prove it. König proved that the cardinality of the continuum cannot be the sum of denumerably many smaller cardinals, and it has been shown that this is the only restriction the accepted axioms of set theory place on its cardinality. Gödel showed that CH was consistent with these axioms and Cohen that its negation was. Together these results prove the independence of CH from the accepted axioms.

Cantor proposed CH in the context of seeking to answer the question ‘What is the identifying nature of continuity?’. These independence results show that, whatever else has been gained from the introduction of transfinite set theory – including greater insight into the import of CH – it has not provided a basis for finally answering this question. This remains the case even when the axioms are supplemented in various plausible ways.