Notes

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 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.



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