Transfinite number

In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, but not necessarily absolutely infinite. 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 formula_1 (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 formula_2 (aleph one) and formula_3. These are called aleph numbers.
Under continuum hypothesis, the number formula_2 is the same as the size of the set of real numbers. This number is sometimes referred to as formula_5 (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 formula_6 is the set of all finite ordinals. It is also the first transfinite ordinal.