The Superiority of Hilbert Arithmetic for Prime Number Theory: I Goldbach’s conjecture proved in Hilbert arithmetic

Goldbach’s conjecture is simply proved in Hilbert arithmetic. However, that proof is either invalid (“incomplete”) or false (“contradictory”) in the standard mathematics obeying Gödel’s objections about the relation of arithmetic to set theory. The proof uses the “apophatic” (holistic) reformulation of the Kochen - Specker theorem and the fundamental randomness of primes in Hilbert arithmetic: both confirmed to be true in previous papers. A few other conjectures, about twin primes, k-twin primes, k-tuple primes (a part of the Hardy - Littlewood conjecture) including about infinite tuples are deduced from Goldbach’s conjecture as corollaries in Hilbert arithmetic. The hypothesis that the asymptotic behaviour predicted by Hardy and Littlewood is false in Hilbert arithmetic is reasoned. A few philosophical observations about “crossing Rubicon to reality” after interpreting the Gleason and Kochen - Specker theorems to the meta-space of human experience to be Hilbert space and considering the opposition of two- versus three-dimensionality meant by them where the former is associated with Modernity and the latter, with Postmodernity: thus, both defined rigorously and mathematically unlike today’s vague humanitarian (cultural and relativistic) research about them and that historical development led to it. Another and quite simple proof of the theorem about the asymptotic limit of the number of primes until any natural number is suggested. The link between Goldbach’s conjecture and Riemann’s hypothesis is demonstrated to be inherent and almost obvious in Hilbert arithmetic. The conclusion explains why the proofs of centuries-old puzzles about primes are so “scandalously” brief and simple in Hilbert arithmetic.