The uniform content of partial and linear orders

Abstract

The principle ADS asserts that every linear order on ω has an infinite ascending or descending sequence. This has been studied extensively in the reverse mathematics literature, beginning with the work of Hirschfeldt and Shore [16]. We introduce the principle ADC, which asserts that every such linear order has an infinite ascending or descending chain. The two are easily seen to be equivalent over the base system RCA 0 of second order arithmetic; they are even computably equivalent. However, we prove that ADC is strictly weaker than ADS under Weihrauch (uniform) reducibility. In fact, we show that even the principle SADS, which is the restriction of ADS to linear orders of type ω + ω * , is not Weihrauch reducible to ADC. In this connection, we define a more natural stable form of ADS that we call General-SADS, which is the restriction of ADS to linear orders of type k + ω, ω + ω * , or ω + k, where k is a finite number. We define General-SADC analogously. We prove that General-SADC is not Weihrauch reducible to SADS, and so in particular, each of SADS and SADC is strictly weaker under Weihrauch reducibility than its general version. Finally, we turn to the principle CAC, which asserts that every partial order on ω has an infinite chain or antichain. This has two previously studied stable variants, SCAC and WSCAC, which were introduced by Hirschfeldt and Jockusch [16], and by Jockusch, Kastermans, Lempp, Lerman, and Solomon [18], respectively, and which are known to be equivalent over RCA 0 . Here, we show that SCAC is strictly weaker than WSCAC under even computable reducibility.

Partial orders and immunity in reverse mathematics

“…The proof of the separation of ADS from CAC is significantly simpler and more modular, as advocated by the author in [20]. Last, we give a simpler separation of two versions of stability for the chain-antichain principles over computable reducibility, which was previously proven by Astor et al [1] by the means of a mutually dependent elaborate notion of forcing. 1…”

“…For every R, S such that ψ(R, S) holds, there exists some X ∈ and someR ⊆ R andS ⊆ S such that ϕ(X ,R,S) holds. By definition of X ∈ ,R ⊆ A 0 orS ⊆ A 1 and therefore either R ⊆ A 0 or S ⊆ A 1 Whenever requiring the sets A 0 and A 1 to be co-c.e., we recover the standard notion of hyperimmunity. Therefore, the restriction of the preservation of dependent hyperimmunity to co-c.e.…”

Ramsey’s theorem and products in the Weihrauch degrees

“…Therefore, there are at most l · 0 + (k − 2l) · 1 + (8 + l − k) · 4 = 32 − 3k + 2l 32 − 2k many collections that satisfy (1). We conclude that there are at most (2k + 8) + (32 − 2k) = 40 bad collections of three fibers.…”

“…We can characterize the bad collections of three fibers: Lemma 5.15. Let k 0 , k 1 2 and let Ψ : k 0 × k 1 → N be a surjective partial function. A collection of three fibers is bad if and only if their union contains either:…”

“…Proof. For part(1), it is enough to show that LPO W wtu-D 2 2 , the rest of the reductions being obvious. This is proved much like Proposition 2.11.…”