*Basic Set Theory (BS)* is the theory in the Language of Set Theory (LST) - $\{\in\}$ - with the following axioms: 1) Extensionality: $\forall x \forall y[\forall z(z \in x \leftrightarrow z \in y) \rightarrow(x=y)]$; 2) Induction schema: $\forall \vec{a}[\forall x((\forall y \in x) \Phi(y, \vec{a}) \rightarrow \Phi(x, \vec{a})) \rightarrow \forall x \Phi(x, \vec{a})]$ where $\Phi$ is any formula of LST with free variables amongst $x, \vec{a}$; 1) Pairing: $\forall x \forall y \exists z \forall w[(w \in z) \leftrightarrow(w=x \vee w=y)]$; 2) Union: $\forall x \exists y \forall z[(z \in y) \leftrightarrow(\exists u \in x)(z \in u)]$; 3) Infinity: $\exists x[\operatorname{On}(x) \wedge(x \neq 0) \wedge(\forall y \in x)(\exists z \in x)(y \in z)]$; 4) Cartesian Product: $\forall x \forall y \exists z \forall u[(u \in z) \leftrightarrow(\exists a \in x)(\exists b \in y)(u=(a, b))]$; 5) $\Sigma_0$-Comprehension (schema): $\forall \vec{a} \forall x \exists y \forall z[(z \in y) \leftrightarrow(z \in x \wedge \Phi(\vec{a}, z))]$, where $\Phi(\vec{a}, z)$ is a $\Sigma_0$ formula of LST.