Robinson arithmetic

Robinson arithmetic is a weak incomplete arithmetic theory.

Definition
Robinson arithmetic has seven axioms on a constant term symbol \(0\) and three function symbols \(S\), \(+\), and \(\cdot\) of arity \(1\), \(2\), and \(2\) respectively. The axioms are:


 * \(Sx \neq 0\) (0 is not the successor of any number.)
 * \((Sx = Sy) \rightarrow x = y\) (If the successor of x is identical to the successor of y, then x and y are identical.)
 * \(y = 0 \lor \exists x (Sx = y)\) (Every number is either 0 or the successor of some number.)
 * \(x + 0 = x\)
 * \(x + Sy = S(x+y)\)
 * \(x \cdot 0 = 0\)
 * \(x \cdot Sy = (x \cdot y) + x\)

Semantics
The standard model \(\mathbb{N}\) of Peano arithmetic forms a model of Robinson arithmetic with respect to the following interpretation:
 * \(0 \in \mathbb{N}\) corresponding to \(0\)
 * Successor function corresponding to \(S\).
 * Addition and multiplication corresponding to \(+\) and \(\cdot\) respectively.

Unlike Peano arithmetic, Robinson arithmetic does not prove the induction schema. In particular, the existence of the maximum element with respect to the ordering \(x < y\) given as \(\exists z(y = x+S(z))\), which is inconsistent with the induction schema, is consistent with Robinson arithmetic. Namely, \(\mathbb{N} \cup \{\)ω|\(\omega\)\(\}\) also forms a model of Robinson arithmetic with respect to a natural extension of the interpretations.