Transfinite Number: Number larger than all finite numbers

These numbers can be categorized into two types: transfinite cardinals and transfinite ordinals.

Transfinite cardinals are infinite numbers used to quantify the different sizes of infinity—of which there are many kinds. For example, the number ${\displaystyle \aleph _{0}}$ (aleph null) is used to refer to the size of the set of natural numbers (or other sets having the same size as the natural numbers). Other such numbers include ${\displaystyle \aleph _{1}}$ (aleph one) and ${\displaystyle \aleph _{2}}$. These are called aleph numbers.

Under continuum hypothesis, the number ${\displaystyle \aleph _{1}}$ is the same as the size of the set of real numbers. This number is sometimes referred to as ${\displaystyle {\mathfrak {c}}}$ (or the cardinality of the continuum).

Aside from transfinite cardinals, transfinite ordinals also exist. These are numbers used to describe the ordering of sets. For example, the ordinal number ${\displaystyle \omega }$ is the set of all finite ordinals. It is also the first transfinite ordinal.

• Infinity