So, is that it? Two types of infinities, countable and uncountable. Well... no. Cantor discovered that there are in fact many infinities of different sizes - infinite infinites in fact. The smallest is \[\aleph_0\] with \[\aleph_0 \lt \aleph_1 \lt \aleph_2 \lt \aleph_3 \lt ... \] However, while Cantor could prove that these infinites existed, he could not prove that the set of reals, that is, uncountable infinity, was necessarily \[\aleph_1\] although he suspected that this was the case.
The Continuum Hypothesis states there is no infinite set with a cardinal number (i.e., cardinality) between that of the infinite set of natural numbers \[\aleph_0 \] and the infinite set of real numbers \[\mathfrak{c}\] That is, \[\mathfrak{c} = \aleph_1 \]
In 1900, David Hilbert made a list of mathematical problems that he deemed most important to solve in the next century. The Continuum Hypothesis made number 1. Some progress was finally made in the mid-20th century, but not in the way anyone expected...
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