[Unprovable statements in] reduced arithmetic

(See page 1152.) Statements that can be proved with induction but are not provable only with Robinson's axioms are: x ≠ Δ x; x + y y + x; x + (y + z) (x + y) + z; 0 + x x; ∃_{x} (Δ x + y z ⇒ y ≠ z); x × y y × x; x × (y × z) (x × y) × z; x × (y + z) x × y + x × z.