Hilbert’s First Problem

Continuum Hypothesis: The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers)

To mathematicians, all infinities are not the same. The infinity of the counting numbers — 1, 2, 3, … — is smaller than the infinity of all the real numbers. And there are towers of still greater infinities beyond the reals. Hilbert’s first problem, also known as the continuum hypothesis, is the statement that there is no infinity in between the infinity of the counting numbers and the infinity of the real numbers. In 1940, Kurt Gödel showed that the continuum hypothesis cannot be proved using the standard axioms of mathematics. In 1963, Paul Cohen showed it cannot be disproved, making the continuum hypothesis independent of the axioms of mathematics.

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