The Experts below are selected from a list of 300 Experts worldwide ranked by ideXlab platform

Koji Nagata - One of the best experts on this subject based on the ideXlab platform.

  • reply to comments on there is no Axiomatic System for the quantum theory
    Journal of Quantum Information Science, 2014
    Co-Authors: Koji Nagata
    Abstract:

    Barros discusses that [Jose Acacio de Barros, Int. J. Theor. Phys. 50, 1828 (2011)] Nagata derives inconsistencies from quantum mechanics [K. Nagata, Int. J. Theor. Phys. 48, 3532 (2009)]. Barros considers that the inconsistencies do not come from quantum mechanics, but from extra assumptions about the reality of observables. Here we discuss the fact that there is a contradiction within the quantum theory. We discuss the fact that only one expected value in a spin-1/2 pure state 〈σx〉rules out the reality of the observable. We do not accept extra assumptions about the reality of observables. We use the actually measured results of quantum measurements (raw data). We use a single Pauli observable. We stress that we can use the quantum theory even if we give up the Axiomatic System for the quantum theory.

  • there is no Axiomatic System for the quantum theory
    International Journal of Theoretical Physics, 2009
    Co-Authors: Koji Nagata
    Abstract:

    We show that there is a contradiction within quantum mechanics. We derive a proposition concerning a quantum expectation value under the assumption of the existence of the directions in a spin-1/2 System. The quantum predictions within the formalism of von Neumann’s projective measurement cannot coexist with the proposition concerning the existence of the directions. Therefore, we have to give up either the existence of the directions or the formalism of von Neumann’s projective measurement. Hence there is a contradiction within the Hilbert space formalism of the quantum theory. This implies that there is no Axiomatic System for the quantum theory. We need new physical theories in order to explain mathematically the handing of raw experimental data.

  • whether quantum mechanics can be almighty even in information science
    arXiv: General Physics, 2008
    Co-Authors: Koji Nagata, Tadao Nakamura
    Abstract:

    We discuss that there is a crucial contradiction within quantum mechanics. We derive a proposition concerning a quantum expectation value under the assumption of the existence of the directions in a spin-1/2 System. The quantum predictions within the formalism of von Neumann's projective measurement cannot coexist with the proposition concerning the existence of the directions. Therefore, we have to give up either the existence of the directions or the formalism of von Neumann's projective measurement. Hence there is a crucial contradiction within the Hilbert space formalism of the quantum theory. This implies that there is no Axiomatic System for the quantum theory. This also reveals that we need new physical theories in order to explain the handing of raw experimental data. We discuss that this crucial contradiction makes the quantum-theoretical formulation of Deutsch's algorithm questionable.

  • There is no Axiomatic System for the quantum theory
    arXiv: General Physics, 2007
    Co-Authors: Koji Nagata
    Abstract:

    Recently, [arXiv:0810.3134] is accepted and published. We derive an inequality with two settings as tests for the existence of the Bloch sphere in a spin-1/2 System. The probability theory of measurement outcome within the formalism of von Neumann projective measurement violates the inequality. Namely, we have to give up the existence of the Bloch sphere. Or, we have to give up the probability theory of measurement outcome within the formalism of von Neumann projective measurement. Hence it turns out that there is a contradiction in the Hilbert space formalism of the quantum theory, viz., there is no Axiomatic System for the theory.

Wensheng Yu - One of the best experts on this subject based on the ideXlab platform.

  • a formal System of Axiomatic set theory in coq
    IEEE Access, 2020
    Co-Authors: Wensheng Yu
    Abstract:

    Formal verification technology has been widely applied in the fields of mathematics and computer science. The formalization of fundamental mathematical theories is particularly essential. Axiomatic set theory is a foundational System of mathematics and has important applications in computer science. Most of the basic concepts and theories in computer science are described and demonstrated in terms of set theory. In this paper, we present a formal System of Axiomatic set theory based on the Coq proof assistant. The Axiomatic System used in the formal System refers to Morse-Kelley set theory which is a relatively complete and concise Axiomatic set theory. In this formal System, we complete the formalization of the basic definitions of sets, functions, ordinal numbers, and cardinal numbers and prove the most commonly used theorems in Coq. Moreover, the non-negative integers are defined, and Peano’s postulates are proved as theorems. According to the axiom of choice, we also present formal proofs of the Hausdorff maximal principle and Schroeder-Bernstein theorem. The whole formalization of the System includes eight axioms, one axiom schema, 62 definitions, and 148 corollaries or theorems. The “Axiomatic set theory” formal System is free from the more apparent paradoxes, and a complete Axiomatic System is constructed through it. It is designed to give a foundation for mathematics quickly and naturally. On the basis of the System, we can prove many famous mathematical theorems and quickly formalize the theories of topology, modern algebra, data structure, database, artificial intelligence, and so on. It will become an essential theoretical basis for mathematics, computer science, philosophy, and other disciplines.

Murat Diker - One of the best experts on this subject based on the ideXlab platform.

  • textures and fuzzy unit operations in rough set theory an approach to fuzzy rough set models
    Fuzzy Sets and Systems, 2017
    Co-Authors: Murat Diker
    Abstract:

    Abstract In this paper, an approach for the fuzzy rough set models is presented using textures and a fuzzy version of the unit operations of Wybraniec-Skardowska. First, a fuzzy unit operation and fuzzy unit co-operation on fuzzy lattices are defined. It is proved that the well-known fuzzy rough set upper approximations as the approximation of Dubois and Prade are fuzzy unit operation. Using fuzzy direlations, an Axiomatic System for fuzzy unit operations is studied and it is shown that fuzzy rough set Systems obtained by different fuzzy logical connectives as Kleene-Dienes and Godel implicators can be generated by the same textural fuzzy direlation. Finally, it is observed that the approximations of two different fuzzy rough set models together constitute two different Galois connections.

Guilong Liu - One of the best experts on this subject based on the ideXlab platform.

  • using one axiom to characterize rough set and fuzzy rough set approximations
    Information Sciences, 2013
    Co-Authors: Guilong Liu
    Abstract:

    Axiomatic approaches are important for understanding the concepts of rough set theory. The properties of the approximation operators constructed in rough set theory are determined by a binary relation. A linkage, in the form of a necessary and sufficient condition, between the constructive approach and an Axiomatic System is established in this paper. Various classes of rough set algebras are characterized by different sets of axioms. With no restriction on the cardinality of the universal set, we use only one axiom to describe the approximation generated by the reflexive, symmetric, and transitive relations, respectively. In addition, we characterize lower and upper approximations of the Pawlak rough set with only one axiom. We also study a similar problem in the context of fuzzy sets.

Anton Zeilinger - One of the best experts on this subject based on the ideXlab platform.

  • logical independence and quantum randomness
    New Journal of Physics, 2010
    Co-Authors: Tomasz Paterek, Peter Klimek, Johannes Kofler, Markus Aspelmeyer, Robert Prevedel, Anton Zeilinger
    Abstract:

    We propose a link between logical independence and quantum physics. We demonstrate that quantum Systems in the eigenstates of Pauli group operators are capable of encoding mathematical axioms and show that Pauli group quantum measurements are capable of revealing whether or not a given proposition is logically dependent on the Axiomatic System. Whenever a mathematical proposition is logically independent of the axioms encoded in the measured state, the measurement associated with the proposition gives random outcomes. This allows for an experimental test of logical independence. Conversely, it also allows for an explanation of the probabilities of random outcomes observed in Pauli group measurements from logical independence without invoking quantum theory. The Axiomatic Systems we study can be completed and are therefore not subject to Godel's incompleteness theorem.

  • mathematical undecidability and quantum randomness
    HASH(0x7f46604f7d98), 2008
    Co-Authors: Tomasz Paterek, Peter Klimek, Caslav Brukner, Johannes Kofler, Robert Prevedel, Anton Zeilinger, Markus Aspelmeyer
    Abstract:

    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. ^