Robinson arithmetic

In mathematics, Robinson arithmetic or Q arithmetic is a primitive axiomatization of number theory. It is essentially Peano arithmetic without the axiom of induction. Therefore, it is often impossible to prove general theorems despite the fact that all special cases can be proven. For example, 5 + 7 = 7 + 5 can be proven in Robinson arithmetic while the general statement that ∀x∀y, x + y = y + x cannot be proven. It was first described in 1950 by R. M. Robinson.

Robinson arithmetic is primarily of interest in the study of Gödel's Incompleteness Theorem because it is a formal system that is considerably weaker than Peano arithmetic, but is one that is still strong enough that Gödel's theorem can still be applied to it.

Axioms

Robinson aritmetic consists of a set called the natural numbers together with a unary function s called the successor function and two binary functions + and *. There is also an element of the set labeled 0. The following axioms then hold:

1. ∀x, s(x) ≠ 0
   • 0 is not the successor of any number...
2. ∀x, x ≠ 0 → ∃y, s(y) = x
   • but all other numbers are successors of some number.
3. ∀x∀y, x ≠ y → s(x) ≠ s(y)
   • If x and y aren't the same, then there successors aren't either.
4. ∀x, x + 0 = x
5. ∀x∀y, x + s(y) = s(x + y)
   • These two axioms together recursively define addition...
6. ∀x, x * 0 = 0
7. ∀x∀y, x * s(y) = (x * y) + y
   • and these two define multiplication.

References

• Robinson, R.M. 1950. "An Essentially Undecidable Axiom System." Proceedings of the Internation Congress of Mathematics, 729-730.
• Swoyer, C. "First Order Theories."