# The Gap of Ignorance: How Uncountable are the Reals?

## Robin Knight

After Cantor provided a method for measuring the sizes of infinite sets, the following question arose: are there any sets of real numbers that are uncountable, so having more elements than the set of rationals, but having fewer elements than the entire set of reals?

Cantor thought that no such sets could exist; his conjecture is the famous Continuum Hypothesis.

Sierpinski, in his 1934 book L’Hypothèse du Continu, summed up 50 years of work by saying that no progress had been made towards finding one. We look at some mathematical and topological reasons why the quest for these sets of intermediate size was so hard, and why mathematicians still can’t agree on whether or not they exist.