nLab Robinson arithmetic

Idea

Robinson arithmetic (after Raphael M. Robinson) is a finitely axiomatized weakening of Peano arithmetic in which the induction schema is dropped and minimal axioms remain.

Definition

Robinson arithmetic, also denoted by Q, is a first-order theory whose signature is that of first-order Peano arithmetic: it consists of a constant $0$, an unary function symbol $s$, and binary function symbols $+, \cdot$. The axioms are

1. $\forall_x \neg s(x) = 0$;

2. $\forall_{x, y} s(x) = s(y) \Rightarrow x = y$;

3. $\forall_x x = 0 \vee \exists_y x = s(y)$;

4. $\forall_x x + 0 = x$,

5. $\forall_{x, y} x + s(y) = s(x+y)$;

6. $\forall_x x \cdot 0 = 0$;

7. $\forall_{x, y} x \cdot s(y) = x \cdot y + x$.

There is no induction scheme.

One sometimes adds axioms for order; however, this is also definable. (Citation in Wikipedia; I can't quite make it work.)

Category of models of Robinson arithmetic

A model of Robinson arithmetic is a set $N$ with an element $0 \in N$, a function $s:N \to N$, and binary operations $(-)+(-):N \times N \to N$ and $(-)\cdot(-):N \times N \to N$, such that

• for all $x \in N$, $\neg s(x) = 0$

• for all $x, y \in N$, $s(x) = s(y)$ implies $x = y$

• for all $x \in N$, $x = 0$ or there exists $y \in N$ where $x = s(y)$

• for all $x \in N$, $x + 0 = x$

• for all $x, y \in N$, $x + s(y) = s(x+y)$

• for all $x \in N$, $x \cdot 0 = 0$

• for all $x, y \in N$, $x \cdot s(y) = x \cdot y + x$

A homomorphism of models of Robinson arithmetic $N$ and $N^{'}$ is a function $f:N \to N^{'}$ such that

• $f(0_N) = 0_{N^{'}}$

• for all $x \in N$, $f(s_N(x)) = s_{N^{'}}(f(x))$

• for all $x, y \in N$, $f(x +_N y) = f(x) +_{N^{'}} f(y)$

• for all $x, y \in N$, $f(x \cdot_N y) = f(x) \cdot_{N^{'}} f(y)$

The category of models of Robinson arithmetic in a universe $\mathcal{U}$ is the category

$$\mathrm{RobinsonArithmetics}_\mathcal{U}$$

whose objects

$$Ob(\mathrm{RobinsonArithmetics}_\mathcal{U})$$

are the models of Nelson arithmetic, and for any two objects

$$A \in Ob(\mathrm{RobinsonArithmetics}_\mathcal{U})$$

and

$$B \in Ob(\mathrm{RobinsonArithmetics}_\mathcal{U})$$

the morphisms

$$Mor_{\mathrm{RobinsonArithmetics}_\mathcal{U}}(A, B)$$

are the homomorphisms of models of Nelson arithmetic as defined above.

This category is an accessible category. If $\mathcal{U}$ satisfies the axiom of finiteness, then the category $\mathrm{RobinsonArithmetics}_\mathcal{U}$ has no initial object and is an empty category.

Theorems

Despite the considerable weakening of what can be proved, the formulas are the same as in Peano arithmetic, and all the background number theory (the Chinese remainder theorem? and the like) needed to develop Gödel codings, incompleteness theorems, and so on is still there. In some sense the axioms are a minimal set needed to carry out this program (and in fact this was in large part the motivation for Robinson).

Models

Unlike the case for Peano arithmetic, system $Q$ admits tractable computable nonstandard? models.

The simplest consists of a single nonstandard number $\infty$ (in addition to all of the standard numbers, which exist in every model), with the rules (where $n$ is a standard number):

• $s(\infty) = \infty$,
• $n + \infty, \infty + n, \infty + \infty = \infty$,
• $0 \cdot \infty, \infty \cdot 0 = 0$, $s(n) \cdot \infty, \infty \cdot s(n), \infty \cdot \infty = \infty$.

See also

• Peano arithmetic
• finite mathematics

References

• Wikipedia, Robinson arithmetic