Figure 2 description

The Turing degrees ${\mathcal{D}}_T$ figure consists of four black circles aligned vertically. The first and lowest is labeled “$\mathbf{0}$ = degree of all computable sets”. The second is labeled “${\mathbf{0}}^{\prime }=\text{deg}(K)=\text{deg}(\mathit{\text{HP}})$”. The third is labeled “${\mathbf{0}}^{″}=\text{deg}(\mathit{\text{TOT}})=\text{deg}(\mathit{\text{FIN}}$)”. Between the third and fourth is a verticle ellipse. The fourth is labeled “ ${\mathbf{0}}^{(n)}$”. The first and second black circles are connected by two pairs of curved lines. The innermost pair is labeled “The c.e. degrees ${\mathcal{E}}_T$”. The outermost pair is labeled “Degrees $a\le {\mathbf{0}}^{\prime }$”.

Figure 3 description

The Arithmetical Hierarchy figure is two vertical parallel lines each has five points with the bottommost one unlabeled; there is a vertical ellipse above the top. The left line’s labeled points are, from bottom to top, ${\mathrm{\Sigma }}_1^0$ through ${\mathrm{\Sigma }}_4^0$. The right line’s labeled points are, from bottom to top, ${\mathrm{\Pi }}_1^0$ through ${\mathrm{\Pi }}_4^0$. Eatch point on the left is connected to the points on the right that are above and below it (e.g., ${\mathrm{\Sigma }}_1^0$ is connected to the bottom unlabeled point on the right and to ${\mathrm{\Pi }}_2^0$). Each of the intersections is also labeled and from bottom to top they are:

• ${\mathrm{\Delta }}_1^0=\text{computable sets}$ for the intersection of the lines from unlabeled to ${\mathrm{\Pi }}_1^0$ and from ${\mathrm{\Sigma }}_1^0$ to unlabeled.
• ${\mathrm{\Delta }}_2^0=\text{sets}{\le }_T{\mathrm{\varnothing }}^{\prime }$ for the intersection of the lines from ${\mathrm{\Sigma }}_1^0$ to ${\mathrm{\Pi }}_2^0$ and ${\mathrm{\Sigma }}_2^0$ to ${\mathrm{\Pi }}_1^0$
• ${\mathrm{\Delta }}_3^0=\text{sets}{\le }_T{\mathrm{\varnothing }}^{″}$ for the intersection of the lines from ${\mathrm{\Sigma }}_2^0$ to ${\mathrm{\Pi }}_3^0$ and ${\mathrm{\Sigma }}_3^0$ to ${\mathrm{\Pi }}_2^0$
• ${\mathrm{\Delta }}_4^0=\text{sets}{\le }_T{\mathrm{\varnothing }}^{‴}$ for the intersection of the lines from ${\mathrm{\Sigma }}_3^0$ to ${\mathrm{\Pi }}_4^0$ and ${\mathrm{\Sigma }}_4^0$ to ${\mathrm{\Pi }}_3^0$

On the left line between the unlabeled bottom point and ${\mathrm{\Sigma }}_1^0$ is the label $K{\equiv }_m{\mathrm{\varnothing }}^{\prime }=$ c.e. sets. Also on the left line between ${\mathrm{\Sigma }}_1^0$ and ${\mathrm{\Sigma }}_2^0$ is the label $\mathit{\text{FIN}}{\equiv }_m{\mathrm{\varnothing }}^{″}$.

On the right line between the unlabeled bottom point and ${\mathrm{\Pi }}_1^0$ is the label = co-c.e. sets $\bar{K}{\equiv }_m\overline{{\mathrm{\varnothing }}^{\prime }}$. Also on the right line between ${\mathrm{\Pi }}_1^0$ and ${\mathrm{\Pi }}_2^0$ is the label $\mathit{\text{TOT}}{\equiv }_m\overline{{\mathrm{\varnothing }}^{″}}$.