On the role of the collection principle for Σ⁰₂-formulas in second-order reverse mathematics

Le résumé fourni par la source

We show that the principle P A R T \mathsf {PART} from Hirschfeldt and Shore is equivalent to the Σ 2 0 \Sigma ^0_2 -Bounding principle B Σ 2 0 B\Sigma ^0_2 over R C A 0 \mathsf {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 D^2_2 implies B Σ 2 0 B\Sigma ^0_2 and is thus indeed equivalent to Stable Ramsey’s Theorem for Pairs ( S R T 2 2 \mathsf {SRT}^2_2 ). This also allows us to conclude that the combinatorial principles I P T 2 2 \mathsf {IPT}^2_2 , S P T 2 2 \mathsf {SPT}^2_2 and S I P T 2 2 \mathsf {SIPT}^2_2 defined by Dzhafarov and Hirst all imply

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