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arXiv:2106.08132

Comments on indeterminism and undecidability

İnanç Şahin

Published Jun 1, 2021• 0 citations
arXivHTMLPDF

Abstract

In a recent paper [1], it has been claimed that the outcomes of a quantum coin toss which is idealized as an infinite binary sequence is 1-random. We also defend the correctness of this claim and assert that the outcomes of quantum measurements can be considered as an infinite 1-random or n-random sequence. In this brief note we present our comments on this claim. We have mostly positive but also some negative comments on the arguments of the paper [1]. Furthermore, we speculate a logical-axiomatic study of nature which we believe can intrinsically provide quantum mechanical probabilities based on 1(n)-randomness.

DOI
10.48550/arxiv.2106.08132

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In a recent paper [1], it has been claimed that the outcomes of a quantum coin toss which is idealized as an infinite binary sequence is 1-random. We also defend the correctness of this claim and assert that the outcomes of quantum measurements can be considered as an infinite 1-random or n-random sequence. In this brief note we present our comments on this claim. We have mostly positive but also some negative comments on the arguments of the paper [1]. Furthermore, we speculate a logical-axiomatic study of nature which we believe can intrinsically provide quantum mechanical probabilities based on 1(n)-randomness.
In a recent paper \cite{Landsman1}, it has been claimed that the outcomes of a quantum coin toss which is idealized as an infinite binary sequence is {\it 1-random}. We also defend the correctness of this claim and assert that the outcomes of quantum measurements can be considered as an infinite {\it 1-random} or {\it n-random} sequence. In this brief note we present our comments on this claim. We have mostly positive but also some negative comments on the arguments of the paper \cite{Landsman1}. Furthermore, we speculate a logical-axiomatic study of nature which we believe can intrinsically provide quantum mechanical probabilities based on {\it 1(n)-randomness}.
The topic of this session is “physical randomness”. It might be doubted whether such a subject exists, for definitions of randomness have hitherto almost all been mathematical in nature. The only exceptions of which I am aware are the preceding paper by Benioff and a paper by Wesley Salmon. These attempts to inject some empirical content into randomness are highly desirable. But anyone attempting to formulate a physically based definition of randomness should at some point make clear what the connection is (if any) with a more traditional notion of disorder - that of indeterminism. Repeated reference to quantum mechanical examples whenever physical randomness is discussed indicates that a primary motivation for considering physical randomness to be important is because of the current belief that data sequences associated with quantum mechanical experiments are irreducibly random. (As an indication of this, in any situation in which physical randomness is discussed, translate the remarks about quantum phenomena into remarks about coin tossing, and they will lose much of their interest.)
We propose a new link between mathematical undecidability and quantum physics. We demonstrate that the states of elementary quantum systems are capable of encoding mathematical axioms and show that quantum measurements are capable of revealing whether a given proposition is decidable or not within the axiomatic system. Whenever a mathematical proposition is undecidable within the axioms encoded in the state, the measurement associated with the proposition gives random outcomes. Our results support the view that quantum randomness is irreducible and a manifestation of mathematical undecidability. Whenever a proposition and a given set of axioms together contain more information than the axioms themselves, the proposition can neither be proved nor disproved from the axioms { it is mathematically undecidable [1, 2]. Here we propose a novel link between mathematical undecidability and quantum physics. We demonstrate that the states of elementary quantum systems are capable of encoding mathematical axioms. Quantum mechanics imposes an upper limit on how much information can be encoded in a quantum state [3, 4], thus limiting the information content of the set of axioms. We show that quantum measurements are capable of revealing whether a given proposition is decidable or not. Whenever a mathematical proposition is undecidable within the system of axioms encoded in the state, the measurement associated with the proposition gives random outcomes. This allows for an experimental test of mathematical undecidability by realizing in the laboratory both the actual quantum states and the required quantum measurements. (To illustrate these ideas, we conducted experiments using the polarization of photons.) Our results support the view that quantum randomness is irreducible [5, 6] and a manifestation of mathematical undecidability. Any formal system is based on axioms, which are propositions that are dened to be true. A proposition is logically independent from a given set of axioms if it can neither be proved nor disproved from the axioms. If a proposition is independent from the axioms, neither the proposition itself nor its negation creates an inconsistency together with the axiomatic system. ^
Quantum mechanics is a physical theory supplying probabilities corresponding to expectation values for measurement outcomes. Indeed, its formalism can be constructed with measurement as a fundamental process, as was done by Schwinger, provided that individual measurements outcomes occur in a random way. The randomness appearing in quantum mechanics, as with other forms of randomness, has often been considered equivalent to a form of indeterminism. Here, it is argued that quantum randomness should instead be understood as a form of unpredictability because, amongst other things, indeterminism is not a necessary condition for randomness. For concreteness, an explication of the randomness of quantum mechanics as the unpredictability of quantum measurement outcomes is provided. Finally, it is shown how this view can be combined with the recently introduced view that the very appearance of individual quantum measurement outcomes can be grounded in the Plenitude principle of Leibniz, a principle variants of which have been utilized in physics by Dirac and Gell‐Mann in relation to the fundamental processes. This move provides further support to Schwinger's “symbolic” derivation of quantum mechanics from measurement.
Abstract Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω , to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.
Abstract Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω, to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.