## Long descriptions for some figures in Recursive Functions

### 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 “$$\mathbf{0}' = \textrm{deg}(K) = \textrm{deg}(\textit{HP})$$”. The third is labeled “$$\mathbf{0}'' = \textrm{deg}(\textit{TOT}) = \textrm{deg}(\textit{FIN}$$)”. 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 $$a \le \mathbf{0}'$$”.

### 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, $$\Sigma^0_1$$ through $$\Sigma^0_4$$. The right line’s labeled points are, from bottom to top, $$\Pi^0_1$$ through $$\Pi^0_4$$. Eatch point on the left is connected to the points on the right that are above and below it (e.g., $$\Sigma^0_1$$ is connected to the bottom unlabeled point on the right and to $$\Pi^0_2$$). Each of the intersections is also labeled and from bottom to top they are:

• $$\Delta^0_1 = \textrm{computable sets}$$ for the intersection of the lines from unlabeled to $$\Pi^0_1$$ and from $$\Sigma^0_1$$ to unlabeled.
• $$\Delta^0_2 = \textrm{sets} \le_T \emptyset'$$ for the intersection of the lines from $$\Sigma^0_1$$ to $$\Pi^0_2$$ and $$\Sigma^0_2$$ to $$\Pi^0_1$$
• $$\Delta^0_3 = \textrm{sets} \le_T \emptyset''$$ for the intersection of the lines from $$\Sigma^0_2$$ to $$\Pi^0_3$$ and $$\Sigma^0_3$$ to $$\Pi^0_2$$
• $$\Delta^0_4 = \textrm{sets} \le_T \emptyset'''$$ for the intersection of the lines from $$\Sigma^0_3$$ to $$\Pi^0_4$$ and $$\Sigma^0_4$$ to $$\Pi^0_3$$

On the the left line between the unlabeled bottom point and $$\Sigma^0_1$$ is the label $$K \equiv_m \emptyset' =$$ c.e. sets. Also on the left line between $$\Sigma^0_1$$ and $$\Sigma^0_2$$ is the label $$\textit{FIN} \equiv_m \emptyset''$$.

On the the right line between the unlabeled bottom point and $$\Pi^0_1$$ is the label = co-c.e. sets $$\overline{K} \equiv_m \overline{\emptyset'}$$. Also on the right line between $$\Pi^0_1$$ and $$\Pi^0_2$$ is the label $$\textit{TOT} \equiv_m \overline{\emptyset''}$$.

This is a file in the archives of the Stanford Encyclopedia of Philosophy.