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  "body": "\n## Definition\n\nLet $\\omega^\\omega$ be the set of [[functions|0XHDTE5YMZ]] from $\\N$ to $\\N$.\nFor $f, g \\in \\omega^\\omega$, write $f \\le^* g$ if $f(n) \\le g(n)$ for all but finitely many $n$.\nLet $\\mathcal{M}$ be the family of [[meagre|0FQPYAY9PJ]] subsets of $\\R$.\nA cardinal characteristic of the continuum is an uncountable [[cardinal|0QKPNZKNEA]] $\\kappa \\le 2^{\\aleph_0}$ of the following kind.\nIt is the least size of a family of reals, or of sets of reals, with a given property.\nThe standard examples include:\n\n1. 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 $f \\in F$.\n2. 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 $f \\in F$.\n3. The additivity $\\operatorname{add}(\\mathcal{M})$ is the least size of a subfamily of $\\mathcal{M}$ whose union is not meagre.\n4. The covering number $\\operatorname{cov}(\\mathcal{M})$ is the least size of a subfamily of $\\mathcal{M}$ whose union is $\\R$.\n5. The uniformity $\\operatorname{non}(\\mathcal{M})$ is the least size of a subset of $\\R$ that is not meagre.\n\n## Remarks\n\nExample: $\\aleph_1 \\le \\mathfrak{b} \\le \\mathfrak{d} \\le \\mathfrak{c}$, where $\\mathfrak{c} = 2^{\\aleph_0}$ is the [[cardinality of the continuum|0WC28W9DNM]].\nCheck: for countably many $f_k$, the function $g(n) = \\max_{k \\le n} f_k(n)$ bounds them all, so $\\mathfrak{b} \u003e \\aleph_0$.\nEach dominating family is unbounded, which gives $\\mathfrak{b} \\le \\mathfrak{d}$, and $\\omega^\\omega$ itself dominates, so $\\mathfrak{d} \\le \\mathfrak{c}$.\nNon-example: the least size of a family of meagre sets whose union is meagre is $1$.\nIt is defined in the same style, but it is not uncountable, so it is not a cardinal characteristic.\nThe same three definitions with the Lebesgue null sets in place of $\\mathcal{M}$ give three more cardinals.\nTogether with $\\mathfrak{b}$, $\\mathfrak{d}$ and the cofinalities of the two ideals, they make up the ten cardinals of Cichoń's diagram.\nUnder the continuum hypothesis all these cardinals equal $\\aleph_1$.\nMartin's axiom implies that they all equal $\\mathfrak{c}$.\n",
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