[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.