Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

[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!= Δx; x+y==y+x; x+(y+z)==(x+y)+z; 0+x==x; Exists[x, (Δx+y==z) \[Implies] y!=z]; x × y == y × x; x × (y × z ) == (x × y) × z; x × (y + z) == x × y + x × z.

From Stephen Wolfram: A New Kind of Science [citation]