ON THE ROLE OF THE COLLECTION PRINCIPLE FOR Σ20-FORMULAS IN SECOND-ORDER REVERSE MATHEMATICS

We show that; the principle PART front Hirschfeldt and Shore is equivalent to the Sigma(0)(2)-Bounding principle B Sigma(0)(2) over RCA(0), answering one of their open questions. Furthermore, we also fill a gap in a proof of Cholak, Jockusch and Slaman by showing that D-2(2) implies B Sigma(0)(2) and is thus indeed equivalent to Stable Ranisey's Theorem for Pairs (SRT22). This also allows us to conclude that the combinatorial principles IPT22, SPT22 and SIPT22 defined by Dzhafarov and Hirst all imply B Sigma(0)(2) and thus that, SPT22 and SIPT22 are both equivalent to SRT22 as well. Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over RCA(0), to the failure of B Sigma(0)(2).