Lecture 16: Constructible Reals
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چکیده
Definition 16.1 (Kleene): A set A ⊆ �� is Σn if there exists an arithmetical formula φ(α,β1, . . . ,βn) such that α ∈ A ⇔ ∃β1∀β2 . . .Qβn φ(α,β1, . . . ,βn) where Q is ∃ if n is odd and Q is ∀ if n is even. Similarly, A⊆ �� is Πn if there exists an arithmetical formula φ(α,β1, . . . ,βn) such that α ∈ A ⇔ ∀β1∃β2 . . .Qβn φ(α,β1, . . . ,βn) where Q is ∀ if n is odd and Q is ∃ if n is even. A set that is Σn and Πn at the same time is called ∆n. A set A is analytical if it is Σ 1 n or Π 1 n.
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