We show how Gödel's first incompleteness theorem has an analog in quantum theory. Gödel's theorem implies endless opportunities for appending axioms to arithmetic, implicitly showing a role for an agent, namely an agent that asserts an axiom. There is an analog of this theorem in physics, to do with the set of explanations of given evidence. We prove that the set of explanations of given evidence is uncountably infinite, thereby showing how contact between theory and experiment depends on activity beyond computation and measurement -- physical activity characterized as agency.
We show how Gödel's first incompleteness theorem has an analog in quantum theory. Gödel's theorem implies endless opportunities for appending axioms to arithmetic, implicitly showing a role for an agent, namely an agent that asserts an axiom. There is an analog of this theorem in physics, to do with the set of explanations of given evidence. We prove that the set of explanations of given evidence is uncountably infinite, thereby showing how contact between theory and experiment depends on activity beyond computation and measurement -- physical activity characterized as agency.
Kurt Gödel’s incompleteness theorem refers to incompleteness in the sense that there are always statements which can be formulated within a formal system, but which re-main unproved within the system. Gödel’s theorem was first connected with the theory of computability by Alan Turing. Until recently, the inconsistency that Gödel’s theo-rem tackle has not been directly shown to correspond to any physical phenomenon. A recent article showed how to apply Gödel’s theorem in the context of many-body quan-tum mechanical calculations. Even so, it is still not known whether it can be applied to the most primitive quantum physical phenomena. In the present study, we used Gödel’s incompleteness theorem in the case of the quantum double-slit experiment by applying a proof of Turing machine’s halting problem. In addition, we made an analogy between Gödel’s theorem (and Church - Turing theses of computability), and quantum compu-ting by extrapolating the term “computation” as low as to the single-quantum level.
Gödel appears to have believed strongly that the human mind cannot be explained in terms of any kind of computational physics, but he remained cautious in formulating this belief as a rigorous consequence of his incompleteness theorems. In this chapter, I discuss a modification of standard Gödel-type logical arguments, these appearing to strengthen Gödel's conclusions, and attempt to provide a persuasive case in support of his standpoint that the actions of the mind must transcend computation. It appears that Gödel did not consider the possibility that the laws of physics might themselves involve noncomputational procedures; accordingly, he found himself driven to the conclusion that mentality must lie beyond the actions of the physical brain. My own arguments, on the other hand, are from the scientific standpoint that the mind is a product of the brain's physical activity. Accordingly, there must be something in the physical actions of the world that itself transcends computation. We do not appear to find such noncomputational action in the known laws of physics, however, so we must seek it in currently undiscovered laws going beyond presently accepted physical theory. I argue that the only plausibly relevant gap in current understanding lies in a fundamental incompleteness in quantum theory, which reveals itself only with significant mass displacements between quantum states (“Schrödinger's cats”). I contend that the need for new physics enters when gravitational effects just begin to play a role. In a scheme developed jointly with Stuart Hameroff, this has direct relevance within neuronal microtubules, and I describe this (still speculative) scheme in the following.
The paper addresses Leon Henkin's proposition as a "lighthouse", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information theory, and quantum mechanics: Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness to be unified and thus reduced as a joint property of infinity and of all infinite sets. However, only Henkin's proposition equivalent to an internal position to infinity is consistent . This can be retraced back to set theory and its axioms, where tha t of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that some essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or "observer'' , or "user" to infinity implied by Henkin's proposition as the only consist ent ones as to infinity. An essential area of contemporary knowledge may be synthesized from a single viewpoint.
The r\^{o}le of observers and observations in physics theories is considered in the light of G\{o}del's incompleteness theorem. Incompleteness arises in G\{o}del's theorem with self-referential propositions, when the system asks to define itself in its own terms. Self-referencing occurs in physics whenever the observer is recognized as being part of the observed system, so the acts of observation are also observables and should become part of the phenomena considered by the theory. This is emphasized by the fact that in many instances the same physical phenomenon may be viewed in more than one way, hence its interpretation depends on the mode of observation. Observations of observations imply a potentially endless hierarchy of levels of observations. Each higher level suggests a larger overview with more profound insight into the universe, empirically implying essentially new discoveries and realizations. New discoveries imply new first principles in the foundation of the theory, so the theory remains open and cannot be complete.
Incompleteness and undecidability theorems have to be revised in view of quantum information and computation theory. qrt.tex 1 As has already been pointed out in Gödel’s centennial paper on the incompleteness af arithmetic [1], the classical undecidability theorems of formal logic [2] and the theory of computable functions [4, 5] are based on semantical pardoxes such as the liar [6] or Richard’s paradox. The method of diagonalization, which was first applied by Cantor for a proof of the undenumerability of real numbers [7], has been applied by Turing for a proof of the recursive undecidability of the halting problem [8]. The halting problem is the problem of whether or not an arbitrary algorithm terminates or produces a particular output and terminates. Assume that the halting problem is decidable. Turing [8] proved that this assumption yields a contradiction. To construct the contradiction, consider
Although Gödel's incompleteness theorem made mathematician recognize that no axiomatic system could completely prove its correctness and that there is an eternal hole between our knowledge and the world, physicists so far continue to work on the approaches based on the hypothesis to completely or approximately know the systems of interest. In this paper, however, I review the recent development of a different approach, a statistical theory based upon the notion of incomplete information. This consideration leads to generalized statistical mechanics characterized by an incompleteness parameter which equals unity when information is complete. The mathematical and physical bases of the information incompleteness are discussed. The application of the concomitant incomplete quantum distribution to correlated electron systems is reviewed.