Mathematical logic and foundations Definition

Robinson arithmetic

Definition

Let $L$ be the signature with a constant symbol $0$ , a unary function symbol $S$ , and binary function symbols $+$ and $\cdot$ . Robinson arithmetic $\mathsf{Q}$ is the first-order theory in $L$ whose axioms are the universal closures of these seven formulas:

$S x \ne 0$ ;
$S x = S y \to x = y$ ;
$x \ne 0 \to \exists y\, (x = S y)$ ;
$x + 0 = x$ ;
$x + S y = S(x + y)$ ;
$x \cdot 0 = 0$ ;
$x \cdot S y = (x \cdot y) + x$ .

Remarks

Example: the natural numbers $\N$ with $0$ , $S(n) = n + 1$ , and the usual addition and multiplication satisfy all seven axioms, so they form a model of $\mathsf{Q}$ .

Non-example: the integers $\Z$ with $S(n) = n + 1$ and the usual operations are not a model of $\mathsf{Q}$ . Axiom 1 fails, because $S(-1) = 0$ .

$\mathsf{Q}$ is the Peano axioms without the induction scheme, with axiom 3 added in its place. $\mathsf{Q}$ does not prove $\forall x\, \forall y\, (x + y = y + x)$ ; there are models of $\mathsf{Q}$ in which addition is not commutative. $\mathsf{Q}$ proves every true sentence of the form $\exists x_1 \cdots \exists x_n\, \varphi$ with $\varphi$ quantifier-free, and every computable function is representable in $\mathsf{Q}$ .

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Peano axioms concept
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Robinson arithmetic definition

Tag	`TT8MXRTVBM`
URI	`m2://TT8MXRTVBM`
Kind	definition
Writing	W0 stub
MSC	03F30
Defines	Robinson arithmetic
Uses	Theory, Signature, Peano axioms
Changed	2 November 2026

Robinson arithmetic

Definition

Remarks

Reading path

Further reading

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