Notions of weak genericity

Abstract

This paper deals with forcing in arithmetic (as first introduced by Feferman [2]) and its connections with recursive function theory. We define for each n ≥ 1 the class of weakly n-generic sets. We prove that these classes merge with the classes of n-generic sets to form the hierarchy suggested by the terminology. Our notation is the same as that of Jockusch [5].

Generics for computable Mathias forcing

“…The following result is the analogue of Theorem 2.3 of Kurtz [9] that every A > T ∅ (n−1) hyperimmune relative to ∅ (n−1) is Turing equivalent to the (n − 1)st jump of a weakly Cohen n-generic real. The proof, although mostly similar, requires a few important modifications.…”

“…(The first half is proved much like its analogue in the Cohen case. See, e.g., [9], Corollary 2.7.) Proposition 2.8.…”

Genericity and UD-random reals

“…The answer is certainly yes for weak 1-generics, since Kurtz has shown that every hyperimmune degree contains a weakly 1-generic real [6,7], and there are Schnorr random reals that are hyperimmune. In fact, even a 1-generic real can be Turing equivalent to a UD-random real, since there is a high 1-generic.…”

Higher Randomness and Genericity

“…Therefore, the sets of generic elements is comeagre. The study of Cohen generics was latter pursued by several authors [Joc80,Kur82,Kur83], by lowering the effective complexity of meagre sets which are used: we do not consider all the meagre sets in a countable model of ZFC, but only some of them. We can for instance keep only the closed sets of empty interior whose complement can be enumerated by a Turing machines.…”

“…For question (1), in [BGM] it was shown that Π Recall that for any lightface pointclass Γ , we say that a sequence G ∈ 2 ω is: We show that the level is precisely Σ 1 1 -genericity. This notion can be considered as a higher analogue of Π 0 1 -genericity, a notion which Jockusch noticed is equivalent to 2-genericity (see [Kur82] and [Kur83]). We also investigate the intermediate notion of Π 1 1 -genericity (the higher analogue of 1-genericity), and consider lowness and cupping questions.…”