# 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 ${\displaystyle \aleph _{0}}$ (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.[1]

Under continuum hypothesis, the number ${\displaystyle \aleph _{1}}$ is the same as the size of the set of real numbers.[2] This number is sometimes referred to as ${\displaystyle {\mathfrak {c}}}$ (or the cardinality of the continuum).[1]

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.[1] It is also the first transfinite ordinal.[3]

## References

1. "Comprehensive List of Set Theory Symbols". Math Vault. 2020-04-11. Retrieved 2020-09-06.
2. "Transfinite number | mathematics". Encyclopedia Britannica. Retrieved 2020-09-06.
3. Weisstein, Eric W. "Ordinal Number". mathworld.wolfram.com. Retrieved 2020-09-06.