On Mathias generic sets

2012/01/29 by Cholak, Peter A., Dzhafarov, Damir D., Hirst, Jeffry L.
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1201.6084

Abstract

We present some results about generics for computable Mathias forcing. The n-generics and weak n-generics in this setting form a strict hierarchy 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 ≥ 3 then it satisfies the jump property G^(n-1) = G' ⊕ ∅⁽ⁿ⁾. We prove that every such G has generalized high degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that G, together with any bi-immune set A ≤_T ∅^(n-1), computes a Cohen n-generic set.

Related

• Generics for Mathias forcing over general Turing ideals
• Relative Definability of n-Generics
• Limits to joining with generics and randoms
• Forcing, genericity and CBERS
• First steps towards a formalization of Forcing