# 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.