  We investigate the strength of a randomness notion $\mathcal R$ as a
set-existence principle in second-order arithmetic: for each $Z$ there is an
$X$ that is $\mathcal R$-random relative to $Z$. We show that the equivalence
between $2$-randomness and being infinitely often $C$-incompressible is
provable in $\mathsf{RCA}_0$. We verify that $\mathsf{RCA}_0$ proves the basic
implications among randomness notions: $2$-random $\Rightarrow$ weakly
$2$-random $\Rightarrow$ Martin-L\"{o}f random $\Rightarrow$ computably random
$\Rightarrow$ Schnorr random. Also, over $\mathsf{RCA}_0$ the existence of
computable randoms is equivalent to the existence of Schnorr randoms. We show
that the existence of balanced randoms is equivalent to the existence of
Martin-L\"{o}f randoms, and we describe a sense in which this result is nearly
optimal.
