---
tag: 66748CE93N
kind: concept
name: Cardinal characteristics of the continuum
msc: ["03E17"]
sources:
  - url: https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum
same_as:
  wikidata: Q17113204
  wikipedia: Cardinal characteristic of the continuum
---

## Summary

Cardinal characteristics of the continuum are cardinals of a special kind.
Each is the least size of a family of reals, or of sets of reals, with some combinatorial property.
Each of them lies between $\aleph_1$ and $\mathfrak{c} = 2^{\aleph_0}$, the [[cardinality of the continuum|0WC28W9DNM]].
Under the [[continuum hypothesis|0MN2MER9WN]] they all equal $\aleph_1$, but without it they can take different values.
The provable inequalities between them form a rich theory.
Standard examples, such as the bounding and dominating numbers, are [[defined|R8S00S3Z94]] separately.

## Explanations

### L1 human

How many meagre sets do you need to cover the real line?
Countably many are never enough, by the Baire category theorem, and continuum many points, each one a meagre set, are always enough.
So the answer is some cardinal between $\aleph_1$ and $2^{\aleph_0}$.
Questions of this type define the cardinal characteristics.
If the continuum hypothesis holds there is nothing in between, but otherwise these numbers can differ, and comparing them is a large part of modern set theory.