How does the continuum hypothesis relate the real numbers to aleph-one?

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 (c).12

In the hierarchy of transfinite numbers developed by Georg Cantor, aleph-null (${\aleph }_0$) represents the cardinality of the infinite set of natural numbers. Cantor proved that the continuum is uncountable and strictly larger than ${\aleph }_0$. The continuum hypothesis asserts that the cardinality of the continuum is exactly ${\aleph }_1$ (aleph-one), which is the smallest uncountable cardinal number.123

Mathematically, this is expressed as:
$$c={\aleph }_1$$

By equating the continuum to ${\aleph }_1$, the hypothesis posits that there is no set with a cardinality intermediate between the natural numbers and the real numbers. This proposition is considered undecidable within the standard Zermelo-Fraenkel axioms of set theory, as it can neither be proven nor disproven using those foundational rules.1234