Robinson Arithmetic - Axioms

Axioms

The background logic of Q is first-order logic with identity, denoted by infix '='. The individuals, called natural numbers, are members of a set called N with a distinguished member 0, called zero. There are three operations over N:

• A unary operation called successor and denoted by prefix S;
• Two binary operations, addition and multiplication, denoted by infix + and by concatenation, respectively.

The following axioms for Q are Q1–Q7 in Burgess (2005: 56), and are also the first seven axioms of second order arithmetic. Variables not bound by an existential quantifier are bound by an implicit universal quantifier.

1. Sx ≠ 0
   • 0 is not the successor of any number.
2. (Sx = Sy) → x = y
   • If the successor of x is identical to the successor of y, then x and y are identical. (1) and (2) yield the minimum of facts about N (it is an infinite set bounded by 0) and S (it is an injective function whose domain is N) needed for non-triviality. The converse of (2) follows from the properties of identity.
3. y=0 ∨ ∃x (Sx = y)
   • Every number is either 0 or the successor of some number. The axiom schema of mathematical induction present in arithmetics stronger than Q turns this axiom into a theorem.
4. x + 0 = x
5. x + Sy = S(x + y)
   • (4) and (5) are the recursive definition of addition.
6. x0 = 0
7. xSy = (xy) + x
   • (6) and (7) are the recursive definition of multiplication.