On the role of the collection principle for ∑2 0 -formulas in second-order reverse mathematics

Abstract

We show that the principle PART from Hirschfeldt and Shore is equivalent to the ∑2 0 -Bounding principle B∑2 0 over RCA0, answering one of their open questions. Furthermore, we also fill a gap in a proof of Cholak, Jockusch and Slaman by showing that D2 2 implies B∑2 0 and is thus indeed equivalent to Stable Ramsey's Theorem for Pairs (SRT2 2 ). This also allows us to conclude that the combinatorial principles IPT 2 2 , SPT2 2 and SIPT2 2 defined by Dzhafarov and Hirst all imply B∑2 0 and thus that SPT2 2 and SIPT2 2 are both equivalent to SRT2 2 as well. Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over RCA0, to the failure of B∑2 0 © 2009 American Mathematical Society.