How does the cardinality of the real numbers compare with aleph-null?

In mathematics, the cardinality of the real numbers, often denoted as c or the continuum, is strictly larger than aleph-null (${\aleph }_0$), which is the cardinality of the set of natural numbers.123

While aleph-null represents the size of countably infinite sets—those that can be placed into a one-to-one correspondence with the natural numbers—the real numbers are uncountable, meaning they cannot be paired with the natural numbers in this way. This distinction was a foundational discovery by Georg Cantor, who proved that not all infinities are equal.1245

The relationship between these two sizes is addressed by the continuum hypothesis, which posits that c is equal to ${\aleph }_1$, the next cardinal number after ${\aleph }_0$, implying there is no set with a cardinality strictly between the natural numbers and the real numbers. However, this hypothesis has been shown to be undecidable within the standard Zermelo-Fraenkel axioms of set theory.1254