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.