We show that for every intermediate \(\Sigma ^0_2\) s-degree (i.e. a nonzero s-degree strictly below the s-degree of the complement of the halting set) there exists an incomparable \(\Pi ^0_1\) s-degree. (The same proof yields a similar result for other positive reducibilities as well, including enumeration reducibility.) As a consequence, for every intermediate \(\Pi ^0_2\) Q-degree (i.e. a nonzero Q-degree strictly below the Q-degree of the halting set) there exists an incomparable \(\Sigma ^0_1\) Q-degree. We also show how these results can be applied to provide proofs or new proofs (essentially already known, although some of them not explicitly noted in the literature) of upper density results in local structures of s-degrees and Q-degrees.

我们表明，对于每个中间\（\ Sigma ^ 0_2 \） s-度（即严格低于停止集合补码s-度的非零s-度），都存在不可比的\（\ Pi ^ 0_1 \）s度 （同样的证明产生其它正面reducibilities类似的结果，以及包括列举还原性。）其结果是，对于每一个中间\（\裨^ 0_2 \）Q -degree（即非零Q -degree严格低于Q -暂停集的度）存在一个无与伦比的\（\ Sigma ^ 0_1 \）Q -度。我们还展示了如何将这些结果用于提供较高密度的证明或新证明（尽管有些文献中未明确指出，但实际上已经知道，尽管其中有些尚未明确指出）会导致s度和Q度的局部结构。