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Notions of weak genericity. (English) Zbl 0549.03042
[Correction of the faulty review Zbl 0542.03028.] The notion of n-generic set was introduced to deal with forcing in arithmetic. A set A is called n-generic iff for every \(\Sigma^ 0_ n\) sentence \(\phi\) of number theory with a constant set, either \(A\Vdash\phi \) or \(A\Vdash\neg \phi\), or iff for every \(\Sigma^ 0_ n\) set \({\mathcal S}\) of strings, there exists a string \(\sigma\subseteq A\) such that either \(\sigma\) is in \({\mathcal S}\) or no extension of \(\sigma\) is in \({\mathcal S}\). It may be generalized along two directions. A is weakly n-forcing if for every \(\Sigma^ 0_ n\) sentence \(\phi\) either \(A\Vdash\neg \neg\phi \) or \(A\Vdash\neg \phi\). A is weakly n-generic if it meets every dense \(\Sigma^ 0_ n\) set \({\mathcal S}\) of strings, i.e., there is a string \(\sigma\in {\mathcal S}\) which is extended by A. In this paper it is shown that every weakly \((n+1)\)-generic set is n-generic and every n-generic set is weakly n-generic and also that if A is weakly n-forcing and weakly n-generic then it is n-generic. A degree of unsolvability is called (weakly) n-generic if it contains a (weakly) n-generic set. A degree a is hyperimmune with respect to a degree b if there is a fixed function \(f\leq a\) which is not dominated by any function recursive in b (here f dominates g iff for all n, f(n)\(\geq g(n))\). The main result of this paper is that a degree b is the n-th jump of a weakly \((n+1)\)-generic degree a iff b is hyperimmune with respect to \(0^{(n)}\) and \(b\geq 0^{(n)}\).
Reviewer: Moh ShawKwei

MSC:
03E40 Other aspects of forcing and Boolean-valued models
03D30 Other degrees and reducibilities in computability and recursion theory
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