RANDOMNESS NOTIONS AND REVERSE MATHEMATICS

We investigate the strength of a randomness notionRas a set-existence principle in second-order arithmetic: for each Z there is an X that is R-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in RCA(0). We verify that RCA(0) proves the basic implications among randomness notions: 2-random double right arrow weakly 2-random double right arrow Martin-Lof random double right arrow computably random double right arrow Schnorr random. Also, over 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-Lof randoms, and we describe a sense in which this result is nearly optimal.