Goldbach Conjecture and First-Order Arithmetic

TL;DR

This paper explores the implications of proving the Goldbach Conjecture within First-Order Arithmetic, suggesting that such a proof would imply the existence of a non-universal proof in a related dynamic system.

Contribution

It introduces a novel framework involving Hyperbolic Classification, Essential Regions, and a Goldbach Conjecture Function to analyze proof validity across different logical systems.

Findings

Proves that a proof of Goldbach in First-Order Arithmetic implies a non-universal proof in an extended system.

Establishes a connection between proof validity and dynamic processes related to natural numbers.

Suggests limitations of formal proof systems in capturing all truths about natural numbers.

Abstract

Using the concepts of Hyperbolic Classification of Natural Numbers, Essential Regions and Goldbach Conjecture Function we prove that the existence of a proof of the Goldbach Conjecture in First-Order Arithmetic would imply the existence of another proof in a certain extension that would not be valid in all states of time associated to natural numbers created by means of adequate dynamic processes.