A purely mathematical proof for the existence of the universe.

Abstract

In this work, I present a minimalist approach to first-order logic and show how it implies an irrefutable equation which connects the field of algorithmic information theory (AIT) to our major theories of physics. The construction of the irrefutable equation produces a new theory of physics where the universe is to be interpreted as the result of a universal Turing machine maximizing the entropy during the calculation of its halting probability, $\Omega$. The construction is sufficiently specific to derive General Relativity and the Dirac equation from pure reason. The minimalist approach is, in many ways, similar to the constructivist project in mathematics but taken to the extreme. The approach starts from first-order logic with no axioms and further removes all rules of inference with the exception of the proof by construction. Although this severely cripples first-order logic it nonetheless gives it the following advantage. From an argument originally made by Plato, I argue that any axioms or rules of inference that are removed increases the of the theory. Taken to the extreme, once all axioms and all rules of inference are removed, the theory becomes entirely irrefutable and specifically in the case of this approach, as the first-order logic system is minimal, the epistemological irrefutability of its theorems is maximal. Using this approach, I construct a universal language defined by a small list of first-order sentences. Each of the sentences claims the existence of an object of language which is provable by construction, the only rule of inference allowed by the minimalist system. As a result of being a theorem of a minimal system, the existence of the constructed universal language is therefore maximally irrefutable. This minimalist method loosely resembles Rene Descartes' derivation of the 'cogito ergo sum', an irrefutable statement, obtained by progressively removing any and all uncertain statements and cataloguing what remains. Like Descartes' method, the minimalist method also produces a statement that cannot be denied. But, unlike Descartes, as the proof is written in the language of first order logic, the irrefutable statement obtained is an actual equation. The equation applies to the group of all statements that cannot be refuted by the application of Descartes' universal doubt method. Part I is the minimalist derivation of the irrefutable equation. The equation obtained is formulated as a Gibbs ensemble and relates the algorithmic notions of provable-sentences to that of entropy. Part III is the derivation of the physical laws. I recover, from the irrefutable equation, the exact mathematical formulation of the major theories of physics; including statistical mechanics, quantum mechanics (QM), special and general relativity (GR) and the holographic principle. These equations are derived entirely from pure reason with no appeal to physical observations. All physical laws obtained are shown to be emergent from an irreducible informational entropy that is associated with the existence of irrefutable statements guaranteed by the minimalist method. Naturally, deriving both the Dirac equation and general relativity from the same theory is highly suggestive that the irrefutable equation should be promoted to a tentative Theory of Everything (ToE)-candidate. This result motivates the ToE claim made in this paper.