Robinson arithmetic

Robinson arithmetic is a weak incomplete arithmetic theory.

Definition
The robinson arithmetic has seven axioms on the number \(0\), the set of natural numbers \(\mathbb{N}\) and three operations over \(\mathbb{N}\):


 * Successor function, denoted by a prefix \(S\).
 * Addition and multiplication, denoted by \(+\) and \(\cdot\) 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.)
 * \(x + 0 = x\)
 * \(x + Sy = S(x+y)\)
 * \(x \cdot 0 = 0\)
 * \(x \cdot Sy = (x \cdot y) + x\)