Abstract:
Quantum Mechanics is the revolutionary theory that changes the structure of human thinking. From Einstein up to now , however , with its enormous successes in the world, there are many scientists that opposed it and denyed it as a complete theory . One of the ways to question the completeness of this theory is to use Gödel’s incompleteness theorems. Kurt Gödel, in his Ph. D. studied and just after working on the Hilbert’s completeness and consistency program in the formal systems, proved two important theorems which are known as the Gödel’s incompleteness theorems . These theorems deny Hilbert’s programs and the completeness of the formal system. Generalizing these theorems to the quantum theory is an over decades challenge in the foundations of physics. In this Paper , we explain the problem and we will discuss and critrize the possible usage of the Gödel’s incompleteness theorems on the quantum mechanics.
Machine summary:
During his doctoral studies and shortly after researching Hilbert's programs of completeness and consistency in formal systems, Kurt Gödel proved two important theorems in logic and mathematics that negate the completeness of any axiomatic theory in arithmetic.
Therefore, the approach of this writing is slightly different from other writings in the humanities, and instead of stating explanatory and general propositions, some details of the theory are dissected; for example, in this writing, the statement of Gödel's incompleteness theorems is provided in his own wording; or the definition of a formal system etc.
Quantum Mechanics and Incompleteness With the definitions provided in previous sections, it can be observed that systems in physics are objects that possess properties; whereas a formal system is a set of axioms whose elements can be physical or even abstract systems; but can quantum mechanics, or theories in physics in general, be considered as a formal system, and can Gödel's theorems be applied to them?
As Svozil states, in the propositions of quantum theory, the issue is not undecidability, but rather uncertainty; for example, in Schrödinger's cat experiment, the discussion is about whether the cat is alive or dead, not that we want to decide whether the cat should stay alive or not; whereas regarding Gödel's theorems in arithmetic and formal machine (computer) systems, it must be able to decide whether to print the proposition if the condition is true.
In fact, as was stated in the section on undecidability, unpredictability, and measurement, in the response to Barak's letter, Gödel did not believe that incompleteness theorems had any connection to physics or quantum mechanics.
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