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What does the continuum hypothesis claim about the real numbers?

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Infinity — infinity Mathematical symbol representing the concept of infinity.

The continuum hypothesis is a statement in set theory regarding the size, or cardinality, of the set of real numbers, often referred to as the continuum.1

Specifically, the hypothesis claims that the cardinality of the continuum is the smallest uncountable cardinal number. In mathematical notation, this is expressed as:1

$2^{ℵ_{0}} = ℵ_{1}$

Here, $ℵ_{0}$ (aleph-null) represents the cardinality of the set of natural numbers, while $ℵ_{1}$ is the next cardinal number in the sequence of transfinite numbers. By asserting this equality, the continuum hypothesis posits that there is no set with a cardinality intermediate between that of the natural numbers and the continuum.12

It is important to note that this hypothesis remains an unresolved problem in mathematics. Work by Kurt Gödel and Paul Cohen demonstrated that the standard Zermelo-Fraenkel axioms of set theory can neither prove nor disprove the continuum hypothesis, leaving its acceptance a subject of ongoing debate within the mathematical community.1