---
tag: 6Q9Z2BM1XR
kind: concept
name: Aleph number
msc: ["03E10"]
specializes: [0EBACQFAZW]
sources:
  - url: https://en.wikipedia.org/wiki/Aleph_number
---

## Summary

The aleph numbers $\aleph_\alpha$, indexed by the [[ordinals|0YSCSX5YZC]], are the infinite [[cardinal numbers|0EBACQFAZW]] of well-orderable sets, listed in increasing order.
The first is [[aleph-null|0V2H5PGJ9B]], the size of the natural numbers, and $\aleph_1$ is the next infinite cardinal.
With the axiom of choice, every infinite set has an aleph as its cardinality, so the alephs measure all infinite sizes.
The [[precise definition|TEWH75TX8Y]] is by transfinite recursion.

## Explanations

### L1 human

The alephs are the sizes of infinity, lined up in order.
The smallest infinite size is $\aleph_0$, the size of $\set{0, 1, 2, \ldots}$.
The next one is $\aleph_1$, then $\aleph_2$, and so on, and after all $\aleph_n$ for finite $n$ comes $\aleph_\omega$.
The real numbers have size $2^{\aleph_0}$, which is at least $\aleph_1$.
Which aleph it is cannot be decided from the usual axioms; the [[continuum hypothesis|0MN2MER9WN]] says it is $\aleph_1$.