Definition 2.2.6 (Language of arithmetic). The language of arithmetic is the first-order language $L_{\text{PA}}$ with signature $(0,1,+,\cdot ,<)$. The base theory of arithmetic is the $L_{\text{PA}}$-theory ${\text{PA}}^{-}$ whose axioms express that:

• (1)
  $+$ and $\cdot$ are commutative and associative, with identity elements $0$ and $1$ respectively;
• (2)
  $\cdot$ distributes over $+$;
• (3)
  $<$ is a linear ordering compatible with $+$ and $\cdot$;
• (4)
  $\forall x.\forall y.(x<y\to \exists z.x+z=y)$;
• (5)
  $0<1\wedge \forall x.(x>0\to x\ge 1)$;
• (6)
  $\forall x.x\ge 0$.

The (first-order) theory of Peano arithmetic PA is obtained from ${\text{PA}}^{-}$ by adding the scheme of induction: for each $L_{\text{PA}}$-formula $\phi (x,\overline{y})$, the axiom