Notes

Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations


Statements in Peano arithmetic

Examples include:

Sqrt[2] is irrational:

Not[Exists[a, Exists[b, b!=0 && a × a == ΔΔ0 × (b × b)]]]

• There are infinitely many primes of the form n^2 + 1:

Not[Exists[n, ForAll[c, Exists[a, Exists[b, (n + c) × (n + c) + Δ0 == ΔΔa × ΔΔb]]]]]

• Every even number (greater than 2) is the sum of two primes (Goldbach's Conjecture; see page 135):

ForAll[a, Exists[b,Exists[c, ΔΔ0 × ΔΔa == b + c && ForAll[d,ForAll[e,ForAll[f,(f==ΔΔd × ΔΔe || f == Δ0) \[Implies] (f != b && f != c)]]]]]]

The last two statements have never been proved true or false, and remain unsolved problems of number theory. The picture shows spacings between n for which n^2+1 is prime.


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