Generics for computable Mathias forcing

Abstract

Abstract. We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak ngenerics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n-generic with n ≥ 2 then it satisfies the jump property. We prove that every such G has generalized high Turing degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real.

Canonical immunity and genericity

“…Several other authors have studied computable Mathias forcing. Notably, Cholak, Dzhafarov, Hirst and Slaman [3] showed that every Mathias generic computes an n-generic. Section 5 below explores the connection between canonical immunity and Mathias genericity.…”

“…Let a ⊆ ω be finite and let A ⊆ ω be an infinite computable set with max(a) < min(A). Together a and A determine a Mathias condition [a, A] given by The poset consisting of all Mathias conditions [a, A] (with A computable) forms the basis of computable Mathias forcing (see Cholak, Dzhafarov, Hirst and Slaman [3]). The key notion in any forcing poset is that of genericity.…”

“…In particular, |D(i Remark. When working with computable Mathias forcing, one approach (see Cholak et al [3]) is to identify each Mathias condition [a, A] with a pair (x, e) where x is a canonical code for the finite set a and e is a Turing machine code for the characteristic function of A. Thus, the statement that [a, A] is a Mathias condition is equivalent to the Π 0 2 statement (1) {e} is total, (2) (∀n)({e}(n) ∈ {0, 1}), (3) (∃ ∞ n)({e}(n) = 1) (i.e., A is infinite) and ( 4) max(a) < min(A).…”

“…In particular, Mathias 3-genericity is sufficient to obtain canonical immunity. Refer to [3] for the definition of Mathias n-genericity.…”

“…In view of the remarks above, Theorem 4.5, along with the result of Theorem 5.1, gives an alternative proof of a result from [3].…”