Cardinal characteristics 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. Under the continuum hypothesis 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 separately.