The arithmetical hierarchy for formulas Second-order arithmetic

a formula called bounded arithmetical, or Δ0, when quantifiers of form ∀n<t or ∃n<t (where n individual variable being quantified , t individual term), where

∀
n
<
t
(
⋯
)


{\displaystyle \forall n<t(\cdots )}

stands for

∀
n
(
n
<
t
→
⋯
)


{\displaystyle \forall n(n<t\rightarrow \cdots )}

and

∃
n
<
t
(
⋯
)


{\displaystyle \exists n<t(\cdots )}

stands for

∃
n
(
n
<
t
∧
⋯
)


{\displaystyle \exists n(n<t\land \cdots )}

.

a formula called Σ1 (or Σ1), respectively Π1 (or Π1) when of form ∃m•(φ), respectively ∀m•(φ) φ bounded arithmetical formula , m individual variable (that free in φ). more generally, formula called Σn, respectively Πn when obtained adding existential, respectively universal, individual quantifiers Πn−1, respectively Σn−1 formula (and Σ0 , Π0 equivalent Δ0). construction, these formulas arithmetical (no class variables ever bound) and, in fact, putting formula in skolem prenex form 1 can see every arithmetical formula equivalent Σn or Πn formula large enough n.