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

Zbl 0542.03028
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