Definition:Arithmetical Hierarchy
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Definition
For every $i \in \N$, $\Sigma_i$ and $\Pi_i$ are sets of well-formed formulae in the language of arithmetic.
They are defined recursively as follows:
Let $\phi$ be logically equivalent to a well-formed formula in the language of arithmetic containing no quantifiers.
Then $\phi \in \Sigma_0$ and $\phi \in \Pi_0$
Let $\phi$ be logically equivalent to $Q_1 Q_2 \dotsm Q_k \psi$ where:
Then $\phi \in \Sigma_{n \mathop + 1}$.
Let $\phi$ be logically equivalent to $Q_1 Q_2 \dotsm Q_k \psi$ where:
- $\psi \in \Sigma_n$
- $Q_j$ is either $\forall x$ or $\exists x < y$, for some variable $x$ and term $y$
Then $\phi \in \Pi_{n \mathop + 1}$.
Sources
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