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+title = "Arithmetic Hierarchy"
+author = "Yann Herklotz"
+tags = []
+categories = []
+backlinks = ["4b1"]
+forwardlinks = []
+zettelid = "4b2"
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+
+This is a hierarchy constructed in the following way, where $k$ is the
+type. $k=0$ is the type of $\mathbb{N}$, whereas $k=1$ gives the type of
+$\mathbb{N}\rightarrow \mathbb{N}$:
+
+-   the formula is $\Sigma^k_{n+1}$ if the outermost quantifier is
+    $\exists^k$, and there are $n$ alternations between blocks of
+    $\exists^k$ and $\forall^k$ at the front of the formula.
+
+-   the formula is $\Pi^k_{n+1}$ if the outermost quantifier is
+    $\forall^k$, and there are $n$ alternations between blocks of
+    $\exists^k$ and $\forall^k$ at the front of the formula.