---
tag: SS6WMEN6KX
kind: concept
name: Beth number
msc: ["03E10"]
specializes: [0EBACQFAZW]
sources:
  - url: https://en.wikipedia.org/wiki/Beth_number
same_as:
  wikidata: Q831092
  wikipedia: Beth number
---

## Summary

The beth numbers $\beth_\alpha$ are the infinite [[cardinal numbers|0EBACQFAZW]] obtained from $\aleph_0$ by taking power sets again and again, and suprema at limit stages.
So $\beth_0 = \aleph_0$, and $\beth_1 = 2^{\aleph_0}$ is the cardinality of the real numbers.
The [[precise definition|ZC4BHVHWE9]] is by transfinite recursion.
The generalized continuum hypothesis says exactly that $\beth_\alpha = \aleph_\alpha$ for every ordinal $\alpha$.

## Explanations

### L1 human

Start with the size of the natural numbers.
The set of all subsets of the natural numbers is strictly bigger, by [[Cantor's theorem|0WTEMGYV2C]]; its size is $\beth_1$, the size of the real line.
The set of all subsets of the real line is bigger still; its size is $\beth_2$, the number of all functions from $\R$ to $\R$.
The beth numbers continue this ladder of power sets forever.