What are transfinite cardinal numbers?

Transfinite cardinal numbers are measures of the size of infinite sets, a concept systematically developed by the German mathematician Georg Cantor in the late 19th century. While finite sets are measured by natural numbers, transfinite cardinals describe collections that are infinite, demonstrating that some infinite sets are larger than others.1234

Key aspects of transfinite cardinals include:

• Hierarchy of Sizes: Cantor proved that the power set of any set is strictly larger than the set itself, leading to an unending hierarchy of transfinite cardinals.12
• Aleph Numbers: These are symbolized by the Hebrew letter aleph ($\aleph$) with subscripts. Aleph-null (${\aleph }_0$) represents the cardinality of countably infinite sets, such as the natural numbers.342
• The Continuum: The cardinality of the real numbers, denoted as c, is larger than ${\aleph }_0$. The continuum hypothesis posits that there is no cardinal number between ${\aleph }_0$ and c (often identified as ${\aleph }_1$).3412
• Arithmetic: Arithmetic with transfinite cardinals differs from finite arithmetic; for instance, ${\aleph }_0+{\aleph }_0={\aleph }_0$ and ${\aleph }_0·{\aleph }_0={\aleph }_0$.42