Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Statements in Peano arithmetic

Examples include:

2 is irrational:

¬ a (b (b 0 a × a (Δ Δ 0) × (b × b)))

• There are infinitely many primes of the form n2 + 1:

¬ n (c (a (b (n + c) × (n + c) + Δ 0 (Δ Δ a) × (Δ Δ b))))

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

a (b (c((Δ Δ 0) × (Δ Δ a) b + c d (e (f ((f (Δ Δ d) × (Δ Δ e) f Δ 0) (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 n2 + 1 is prime.

Statements in Peano arithmetic image 1

Image Source Notebooks:

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