Mathematical definition of \(\Delta_0\)-formulas

Definition

1. If \(P \) is a predicate symbol, \(P \in \sigma \), and \( 𝑡_1,…,t_n \) are symbols of constants of the signature \(\sigma\), or variables, then \( P(𝑡_1,…,t_n)\) is a \(\Delta_0 \)-formula.
2. If \( \phi \) and \( \psi \) are \( \Delta_0 \)-formulas, then \( \phi \wedge \psi, \phi \vee \psi , \phi \rightarrow \psi, \neg \psi \) are \( \Delta_0 \)-formulas.
3. If \( \phi \) is a \( \Delta_0 \)-formula, 𝑥 is a variable, and 𝑙 is a finite list, then \( (\forall x \in l)\phi (x) \) and \( (\exists x \in l)\phi (x) \) are \( \Delta_0 \)-formulas.
4. There are no other \( \Delta_0 \)-formulas.