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:

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. For example, $$\mathbb{N} \cup \{$$$$\omega$$$$\}$$ also forms a model of Robinson arithmetic with respect to a natural extension of the interpretations.