Equivalence of COH and CADS over RCA_0

Establish whether the cohesiveness principle COH and the cohesive ascending/descending sequence principle CADS are equivalent over RCA_0.

Background

COH states that every countable sequence of sets has an infinite cohesive set, while CADS states that every infinite linear order has a suborder of type ω, ω* or ω+ω*. Hirschfeldt and Shore proved that COH implies CADS over RCA_0 and that the converse follows over RCA_0+B_2.

The paper notes that the equivalence over the base theory RCA_0 itself was left unresolved, even though COH and CADS are computably equivalent as problems.

References

Hirschfeldt and Shore proved in particular that $COH$ implies $CADS$ over~$RCA_0$ and that the converse holds over~$RCA_0 + B_2$, but left the equivalence over~$RCA_0$ open.

— The weakness of typicality  (2609.00884 - Astor et al., 1 Sep 2026) in Section 7.3, subsection “Cohesiveness principles”