Cone avoiding closed sets

Abstract

We prove that for an arbitrary subtree T of 2 <ω with each element extendable to a path, a given countable class M closed under disjoint union, and any set A, if none of the members of M strongly k-enumerate T for any k, then there exists an infinite set contained in either A orĀ such that for every C ∈ M, C ⊕ G also does not strongly k-enumerate T . We give applications of this result, which include: (1) RT 2 2 doesn't imply WWKL 0 ; (2) [Ambos-Spies et al. [1]] DNR is strictly weaker than WWKL 0 ; (3) ] for any Martin-Löf random set A either A orĀ contains an infinite subset that does not compute any Martin-Löf random set; etc. We also discuss further generalizations of this result.2010 Mathematics Subject Classification. Primary 03B30; Secondary 03F35 03C62 68Q30 03D32 03D80 28A78.

Extracting randomness within a subset is hard

“…The result is interesting because it seems that the combinatorial argument in [8] relies heavily on the fact that A and A form a partition of ω.…”

Partial orders and immunity in reverse mathematics

“…We can easily relate the notion of preservation of c.b-immunity with the existing notion of constant-bound enumeration avoidance defined by Liu [18] to separate RT 2 2 from WWKL over RCA 0 . Proof.…”

The reverse mathematics of non-decreasing subsequences

“…It can be seen as asserting the existence of a random real, in the sense of Martin-Löf [4]. Liu [10] introduced the notion of constant-bound enumeration avoidance to separate Ramsey's theorem for pairs from weak weak König's lemma. We shall use the same notion to separate CNS from WWKL.…”

“…Pick any x ∈ X. By iteratively applying strong c.b-enum avoidance of the infinite pigeonhole principle [10], define a finite sequence X = X 0 ⊇ X 1 ⊇ · · · ⊇ X k of infinite sets such that for each i < k, C has no X i+1 ⊕ C-computable c.b-enum and either there is some n < g i (x) such that g i (y) = n for each y ∈ X i+1 , or g i (y) ≥ g i (x) for each y ∈ X i+1 . If we are in the former case for some i < k, then the condition d = (F, X i+1 , S) is an extension of c such that #(d) < #(c).…”