Why is the cardinality of the real numbers larger than 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.12

This distinction was established by the German mathematician Georg Cantor in the late 19th century. Cantor proved that while the set of natural numbers is countably infinite (denumerable), the set of real numbers is uncountable. He demonstrated this using his diagonalization argument, which shows that it is impossible to create a one-to-one correspondence between the natural numbers and the real numbers. Because the real numbers cannot be paired with the natural numbers, they represent a "larger" infinity.1345

This discovery led to the development of transfinite numbers, which classify the sizes of infinite sets. The continuum hypothesis further posits that there is no set with a cardinality between ${\aleph }_0$ and the continuum, though this remains an undecidable proposition in standard set theory.123