---
tag: R8S00S3Z94
kind: definition
name: Cardinal characteristics of the continuum
msc: ["03E17"]
defines: 66748CE93N
uses: [0QKPNZKNEA, 0WC28W9DNM, 0FQPYAY9PJ, 0XHDTE5YMZ]
sources:
  - url: https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum
---

## Definition

Let $\omega^\omega$ be the set of [[functions|0XHDTE5YMZ]] from $\N$ to $\N$.
For $f, g \in \omega^\omega$, write $f \le^* g$ if $f(n) \le g(n)$ for all but finitely many $n$.
Let $\mathcal{M}$ be the family of [[meagre|0FQPYAY9PJ]] subsets of $\R$.
A cardinal characteristic of the continuum is an uncountable [[cardinal|0QKPNZKNEA]] $\kappa \le 2^{\aleph_0}$ of the following kind.
It is the least size of a family of reals, or of sets of reals, with a given property.
The standard examples include:

1. The bounding number $\mathfrak{b}$ is the least size of a set $F \subseteq \omega^\omega$ with no $g$ such that $f \le^* g$ for all $f \in F$.
2. The dominating number $\mathfrak{d}$ is the least size of a set $F \subseteq \omega^\omega$ such that each $g \in \omega^\omega$ has $g \le^* f$ for some $f \in F$.
3. The additivity $\operatorname{add}(\mathcal{M})$ is the least size of a subfamily of $\mathcal{M}$ whose union is not meagre.
4. The covering number $\operatorname{cov}(\mathcal{M})$ is the least size of a subfamily of $\mathcal{M}$ whose union is $\R$.
5. The uniformity $\operatorname{non}(\mathcal{M})$ is the least size of a subset of $\R$ that is not meagre.

## Remarks

Example: $\aleph_1 \le \mathfrak{b} \le \mathfrak{d} \le \mathfrak{c}$, where $\mathfrak{c} = 2^{\aleph_0}$ is the [[cardinality of the continuum|0WC28W9DNM]].
Check: for countably many $f_k$, the function $g(n) = \max_{k \le n} f_k(n)$ bounds them all, so $\mathfrak{b} > \aleph_0$.
Each dominating family is unbounded, which gives $\mathfrak{b} \le \mathfrak{d}$, and $\omega^\omega$ itself dominates, so $\mathfrak{d} \le \mathfrak{c}$.
Non-example: the least size of a family of meagre sets whose union is meagre is $1$.
It is defined in the same style, but it is not uncountable, so it is not a cardinal characteristic.
The same three definitions with the Lebesgue null sets in place of $\mathcal{M}$ give three more cardinals.
Together with $\mathfrak{b}$, $\mathfrak{d}$ and the cofinalities of the two ideals, they make up the ten cardinals of Cichoń's diagram.
Under the continuum hypothesis all these cardinals equal $\aleph_1$.
Martin's axiom implies that they all equal $\mathfrak{c}$.