In the late 1980s, Selivanov used typed Boolean combinations of arithmetical sets to extend the Ershov hierarchy beyond ${\mathrm{\Delta }}_2^0$ sets. Similar to the Ershov hierarchy, Selivanov's fine hierarchy ${\{{\mathrm{\Sigma }}_{\gamma }\}}_{\gamma <{\epsilon }_0}$ proceeds through transfinite levels below ${\epsilon }_0$ to cover all arithmetical sets. In this paper we use a 0‴ construction to show that the ${\mathrm{\Sigma }}_3^0$ Turing degrees are properly contained in the ${\mathrm{\Sigma }}_{{\omega }^{\omega }+2}$ Turing degrees (to be defined); intuitively, the latter class consists of “non-uniformly ${\mathrm{\Sigma }}_3^0$ sets” in the sense that will be clarified in the introduction. The question whether the hierarchy was proper at this level with respect to Turing reducibility remained open for over 20 years.

在1980年代后期，Selivanov使用算术集的类型化布尔组合将Ershov层次结构扩展到 ${\mathrm{\Delta }}_2^0$套。Selivanov的精细层次类似于Ershov层次 ${\{{\mathrm{\Sigma }}_{\gamma }\}}_{\gamma <{\epsilon }_0}$ 通过下面的超限水平进行 ${\epsilon }_0$涵盖所有算术集。在本文中，我们使用0‴构造来表明${\mathrm{\Sigma }}_3^0$图灵度正确包含在${\mathrm{\Sigma }}_{{\omega }^{\omega }+2}$图灵度（待定）；从直觉上讲，后一类包括“${\mathrm{\Sigma }}_3^0$设置”，其含义将在导言中阐明。关于图灵可简化性，层次结构在此级别上是否合适的问题已有20多年的历史了。