1. The document discusses infinity and the axiom of choice in mathematics. It describes how mathematics approaches infinity through axioms, negation of axioms, and postulating properties of infinite sets by analogy to finite sets.
2. It explains formulations of the axiom of choice like Zorn's lemma and defines paradoxes like Banach-Tarski that rely on the axiom of choice. It also discusses the continuum hypothesis.
3. The author argues that quantum mechanics provides empirical evidence supporting the axiom of choice since entanglement is analogous to the Banach-Tarski paradox. They also discuss negating the continuum hypothesis and axiom of foundation.
This document provides an outline of string theory. It begins with background on reductionism in physics and the unification of forces. String theory emerged as a way to address difficulties in quantizing gravity. There are five consistent string theories in 10 dimensions: type I open superstring theory with oriented strings; type IIA closed superstring theory with two independent sets of supersymmetry; heterotic string theories that combine bosonic and supersymmetric strings. String theory led to the discovery of supersymmetry and relates fundamental forces and particles to vibrational modes of strings.
Against bounded-indefinite-extensibilityjamesstudd
This document discusses absolutism and relativism about quantifiers. It summarizes an argument by Dummett for relativism and considers two options for interpreting Dummett's use of "definite totality." Option 1, taking it to mean a set-like object, fails because the conclusion is not a threat to absolutism. Option 2, taking it to mean a "plurality," also fails because Dummett's premise is inconsistent in plural logic. The document then considers regimenting the paradoxes in a sorted logic to better state relativism and avoid trivial objections. It proposes treating interpretations as sequences and using auxiliary assumptions to formulate a paradox.
This document provides an introduction to general relativity. It begins by summarizing the key aspects of special relativity, including that spacetime is four-dimensional and transformations between inertial frames form the Poincare group. It then discusses the equivalence principle and introduces curved coordinates to describe gravity. The document derives the affine connection and Riemann curvature tensor, and introduces the metric tensor. It provides the perturbative expansion leading to Einstein's field equations and discusses solutions like the Schwarzschild metric and gravitational radiation.
This document summarizes and compares different models of a beam balance experiment. It finds that simpler models are often nested within more complex models. The simplest model (SIMP) represents the mass ratio as a single parameter, while a more complex model (COMP) adds a second parameter. An even more complex model (COMPTOO) adds a third object on the beam. The document examines problems with selecting the best model based only on fit to the data, and discusses alternatives like parsimony and falsification.
Three levels of scientific hypotheses are curves/functions, models, and theories. Curves represent empirical relationships between variables, models incorporate adjustable parameters, and theories are broad sets of principles. Curve fitting involves three steps: 1) determining variables, 2) selecting a model family of curves, and 3) estimating parameter values for the best fitting curve. Models play an essential role by representing families of curves and allowing indirect confirmation of hypotheses. Overlooking models can misconstrue processes like measurement and curve fitting.
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics
Then the proved and even experimentally confirmed completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics
Infinity is equivalent to a second and independent finiteness
Two independent Peano arithmetics as well as one single Hilbert space as an unification of geometry and arithmetic are sufficient to the self-foundation of mathematics
Quantum mechanics is inseparable from the foundation of mathematics and thus from set theory particularly
In this talk, I looked at three theorems due to Poincaré concerning the long-term behavior of dynamical systems, namely, the Poincaré–Perron Theorem, the Poincaré–Bendixson Theorem and the rather counterintuitive Poincaré Recurrence Theorem, along with their proofs. Certain aspects of ergodic theory and topological dynamics, for instance, their applications in number theory and analysis (Green–Tao Theorem and Szemerédi’s Theorem), were also touched upon.
This document provides an outline of string theory. It begins with background on reductionism in physics and the unification of forces. String theory emerged as a way to address difficulties in quantizing gravity. There are five consistent string theories in 10 dimensions: type I open superstring theory with oriented strings; type IIA closed superstring theory with two independent sets of supersymmetry; heterotic string theories that combine bosonic and supersymmetric strings. String theory led to the discovery of supersymmetry and relates fundamental forces and particles to vibrational modes of strings.
Against bounded-indefinite-extensibilityjamesstudd
This document discusses absolutism and relativism about quantifiers. It summarizes an argument by Dummett for relativism and considers two options for interpreting Dummett's use of "definite totality." Option 1, taking it to mean a set-like object, fails because the conclusion is not a threat to absolutism. Option 2, taking it to mean a "plurality," also fails because Dummett's premise is inconsistent in plural logic. The document then considers regimenting the paradoxes in a sorted logic to better state relativism and avoid trivial objections. It proposes treating interpretations as sequences and using auxiliary assumptions to formulate a paradox.
This document provides an introduction to general relativity. It begins by summarizing the key aspects of special relativity, including that spacetime is four-dimensional and transformations between inertial frames form the Poincare group. It then discusses the equivalence principle and introduces curved coordinates to describe gravity. The document derives the affine connection and Riemann curvature tensor, and introduces the metric tensor. It provides the perturbative expansion leading to Einstein's field equations and discusses solutions like the Schwarzschild metric and gravitational radiation.
This document summarizes and compares different models of a beam balance experiment. It finds that simpler models are often nested within more complex models. The simplest model (SIMP) represents the mass ratio as a single parameter, while a more complex model (COMP) adds a second parameter. An even more complex model (COMPTOO) adds a third object on the beam. The document examines problems with selecting the best model based only on fit to the data, and discusses alternatives like parsimony and falsification.
Three levels of scientific hypotheses are curves/functions, models, and theories. Curves represent empirical relationships between variables, models incorporate adjustable parameters, and theories are broad sets of principles. Curve fitting involves three steps: 1) determining variables, 2) selecting a model family of curves, and 3) estimating parameter values for the best fitting curve. Models play an essential role by representing families of curves and allowing indirect confirmation of hypotheses. Overlooking models can misconstrue processes like measurement and curve fitting.
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics
Then the proved and even experimentally confirmed completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics
Infinity is equivalent to a second and independent finiteness
Two independent Peano arithmetics as well as one single Hilbert space as an unification of geometry and arithmetic are sufficient to the self-foundation of mathematics
Quantum mechanics is inseparable from the foundation of mathematics and thus from set theory particularly
In this talk, I looked at three theorems due to Poincaré concerning the long-term behavior of dynamical systems, namely, the Poincaré–Perron Theorem, the Poincaré–Bendixson Theorem and the rather counterintuitive Poincaré Recurrence Theorem, along with their proofs. Certain aspects of ergodic theory and topological dynamics, for instance, their applications in number theory and analysis (Green–Tao Theorem and Szemerédi’s Theorem), were also touched upon.
Indeterminism in quantum mechanics can be interpreted as a form of occasionalism, which is called “quantum occasionalism”
It in turn implies the option for any probabilistic determinism and thus for causal determinism as a particular case to be extended equivalently to that form of occasionalism
Problem of the direct quantum-information transformation of chemical substanceVasil Penchev
1. The document discusses the possibility of directly transforming one chemical substance into another through a "Trigger field" as proposed in a science fiction novel.
2. It explores how quantum mechanics, which underlies chemistry, can be interpreted in terms of quantum information and entanglement. Entanglement could theoretically allow the direct alteration of a substance's quantum information and transformation into another substance from a distance.
3. While a standalone "Trigger field" is not currently known to exist, the document argues that entanglement provides a theoretical framework for how a field could directly change a substance's quantum information and transform it into another, as envisioned in the science fiction story.
PRINCETON'S SPIRIT
Contemporary neopythagoreanism – The lodged at Princeton refugees – On quantum information as a mathematical doctrine – “The sixth problem” of Hilbert– Axiomatic logics, geometries, but why not also “physicses”? – The axiomatizing both of the theory of probability and of mechanics – The coincidence of model and reality as a solution of “the sixth problem” of Hilbert – The theorem about the absence of hidden parameters as a proof for the coincidence of model and reality – Bell’s inequalities as a generalization of von Neumann’s theorem – “The second problem” of Hilbert – Why “arithmetization”? – Arithmetization vs. geometrization? – Meta-mathematics: the foundation or self-foundation of mathematics – The problem of actual infinity – Actual infinity as a derivative of wholeness – The theory of Hilbert space as that domain of mathematics, which is able to found itself – Mathematical existence and existence in general – Mathematics as ontology: Pythagoreanism – Completeness, consistency … and additivity – The quantum nostrum of non-additivity – Transfinite induction: Peano or Gentzen arithmetic – A dual foundation of arithmetic: the “geometrization” of arithmetic – Gödel and Hilbert mathematics – The Kochen and Specker theorem – “Hidden parameter” does not “the element of reality” – The theorem of Kochen and Specker as a generalization of von Neumann’s – Duality, holism, and numberness (numericality) – Of I Ching generating Yin and Yang – The cyclic and holistic paradigm of dualistic Pythagoreanism versus the classical bipolar episteme – Any complete and consistent structure is non-additive − The incompleteness both of quantum mechanics and arithmetic? – Choice, number, and probability − Ψ-function in a generalized notation – The sense of Einstein’s “common covariance” – “Princeton” also for gauge theories – More about “dualistic pythagoreanism” – Quantity and property – Projection operator as statement (à la von Neumann)− Simultaneous undecidability – Does the notion of physical quantity imply the invariance of time moments? – Commuting and non-commuting operators – Perfecting the notion for simultaneous immeasurability – Quantum mechanics in Procrustean bed – The world is also a mathematical structure for its essence
Quantum Measure from a Philosophical ViewpointVasil Penchev
1. The paper discusses quantum measure from a philosophical viewpoint. Quantum measure is a proposed three-dimensional measure that can measure both continuous and discrete phenomena uniformly.
2. Quantum measure is proposed as a more universal measure than Lebesgue or Borel measures. It is constructed to be invariant to the axiom of choice and resolves issues like how probability can become physical quantity.
3. The relationship between quantum mechanics and general relativity can be understood through this framework. They are complementary aspects of the quantum state of a system like the universe. However, general relativity is built assuming the axiom of choice while quantum mechanics is invariant to it, posing challenges for a unified theory of quantum gravity.
Quantum information as the information of infinite seriesVasil Penchev
The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The qubit, can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question.
Gödel’s completeness (1930) nd incompleteness (1931) theorems: A new reading ...Vasil Penchev
(T2) Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of inf(T1) Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation
Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity
That is nothing else than right a new reading at issue and comparative interpretation of Gödel’s papers meant here
inity
The most utilized example of those generalizations is the separable complex Hilbert space: it is able to equate the possibility of pure existence to the probability of statistical ensemble
(T3) Any generalization of Peano arithmetic consistent to infinity, e.g. the separable complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself
Action as the common measure of randomness and free willVasil Penchev
One needs a common measure of randomness and free well:
Both cause results, which are comparable
Given any fact without being known its origin, the question whether its origin is natural (i.e. random, occasional) or freewill action (intentional, constructive) arises
The result in both cases is one and the same as well as action and even the physical quantity of action
However the natural process being just random would require much more time in comparison with any intentional action for one and the same result
History as the ontology of time requires to be understood what time is
Time is the transformation of future into the past by the choices in the present
History should be grounded on that understanding of historical time, which would include the present and future rather than only the past
The thesis is: Einstein, Podolsky and Rosen’s argument (1935, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? ) is another interpretation of the famous Gödel incompleteness argument (1931, Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I ) in terms of quantum mechanics
Representation & Realityby Language (How to make a home quantum computer) Vasil Penchev
Reality as if is doubled in relation to language
We will model this doubling by two Turing machines (i.e. by usual computers) in a kind of “dialog”: the one for reality, the other for its image in language
The two ones have to reach the state of equilibrium to each other
At last, one can demonstrate that the pair of them is equivalent to a quantum computer
One can construct a model of two independent Turing machines allowing of a series of relevant interpretations:
Language
Quantum computer
Representation and metaphor
Reality and ontology
In turn that model is based on the concepts of choice and information
Математизирането на историята: число и битиеVasil Penchev
Книгата е първа част от по-широк замисъл − „Числа“. Така е назована на гръцки и от него на всички езици четвърта глава от Библията, посветена на преброяването на изра-илтяните и похода в пустинята. Стремежът е се да представи едно осъвременено пита-горейство. Предвиждат се още две части:
Синтактично-семантично интерпретиране на вълновата функция. Число и знак
Математизирането на историята. Число и история
Книгата е предназначена за научни работници в областта на философията, историята и математиката,
This document discusses quantum computers from a mathematical perspective by comparing them to Turing machines. It proposes that a quantum computer can be modeled as a Turing machine with an infinite tape of "qubits" rather than bits. This raises philosophical questions about the relationship between mathematical models and reality when dealing with infinity. The document also explores how concepts like information, choice, and measurement are understood differently in quantum as opposed to classical computation.
Quantum Mechanics as a Measure Theory: The Theory of Quantum MeasureVasil Penchev
This document discusses representing quantum mechanics as a measure theory, with the key points being:
1. Quantum measure can unify the measurement of discrete and continuous quantities by treating them as "much" and "many".
2. The unit of quantum measure is the qubit, allowing it to jointly measure probability, quantity, order, and disorder.
3. All physical processes can be interpreted as computations of a quantum computer, with the universal substance being quantum information.
4. Quantum measure can provide a nonlocal explanation for the Aharonov-Bohm effect by linking it to the electromagnetic nature of space-time itself.
The question is:
•
How should skepticism refer to itself?
•
The classical example might be the doubt of Descartes, which led him to the necessary obviousness of who doubts
•
The formal logical structure is the same as the “antinomy of the Liar”
•
That new interpretation of it can be called “antinomy of the Skeptic”
The document discusses interpretations of quantum mechanics including the Copenhagen interpretation, many-worlds interpretation, and transactional interpretation. It summarizes four quantum paradoxes around wave-particle duality, quantum measurement, and non-locality. It then provides more details on the transactional interpretation, explaining how it uses advanced and retarded waves to describe quantum events and resolve the paradoxes without needing observers or wavefunction collapse. Finally, it discusses how interpretations cannot be experimentally tested but notes one potential exception with a new experiment.
The document discusses quantum mechanics and three interpretations of its formalism: the Copenhagen interpretation, the many-worlds interpretation, and the transactional interpretation. It describes four quantum paradoxes around non-locality, wave-particle duality, and wave function collapse. Each interpretation aims to resolve these paradoxes while linking the mathematical formalism to physical phenomena.
"Possible Worlds and Substances“ by Vladislav TerekhovichVasil Penchev
The document discusses unifying the categories of possible worlds and substances through the concept of possible histories. It suggests describing the set of possible histories metaphysically as the gradual development of possible substances into distinguishable possible worlds.
The comment aims to mention related philosophical and scientific ideas, including unifying Gibbs and Boltzmann thermodynamics, describing discrete and continuous motions in quantum mechanics, and interpreting quantum mechanics' mathematical formalism through various metaphysical lenses.
It argues these ideas are members of the same "family" connected by a tendency toward mathematics. This could realize a form of Pythagoreanism identifying mathematics and reality, resolving problems in the conclusion of the original paper by identifying possibility as a universal probability substance underlying
Indeterminism in quantum mechanics can be interpreted as a form of occasionalism, which is called “quantum occasionalism”
It in turn implies the option for any probabilistic determinism and thus for causal determinism as a particular case to be extended equivalently to that form of occasionalism
Problem of the direct quantum-information transformation of chemical substanceVasil Penchev
1. The document discusses the possibility of directly transforming one chemical substance into another through a "Trigger field" as proposed in a science fiction novel.
2. It explores how quantum mechanics, which underlies chemistry, can be interpreted in terms of quantum information and entanglement. Entanglement could theoretically allow the direct alteration of a substance's quantum information and transformation into another substance from a distance.
3. While a standalone "Trigger field" is not currently known to exist, the document argues that entanglement provides a theoretical framework for how a field could directly change a substance's quantum information and transform it into another, as envisioned in the science fiction story.
PRINCETON'S SPIRIT
Contemporary neopythagoreanism – The lodged at Princeton refugees – On quantum information as a mathematical doctrine – “The sixth problem” of Hilbert– Axiomatic logics, geometries, but why not also “physicses”? – The axiomatizing both of the theory of probability and of mechanics – The coincidence of model and reality as a solution of “the sixth problem” of Hilbert – The theorem about the absence of hidden parameters as a proof for the coincidence of model and reality – Bell’s inequalities as a generalization of von Neumann’s theorem – “The second problem” of Hilbert – Why “arithmetization”? – Arithmetization vs. geometrization? – Meta-mathematics: the foundation or self-foundation of mathematics – The problem of actual infinity – Actual infinity as a derivative of wholeness – The theory of Hilbert space as that domain of mathematics, which is able to found itself – Mathematical existence and existence in general – Mathematics as ontology: Pythagoreanism – Completeness, consistency … and additivity – The quantum nostrum of non-additivity – Transfinite induction: Peano or Gentzen arithmetic – A dual foundation of arithmetic: the “geometrization” of arithmetic – Gödel and Hilbert mathematics – The Kochen and Specker theorem – “Hidden parameter” does not “the element of reality” – The theorem of Kochen and Specker as a generalization of von Neumann’s – Duality, holism, and numberness (numericality) – Of I Ching generating Yin and Yang – The cyclic and holistic paradigm of dualistic Pythagoreanism versus the classical bipolar episteme – Any complete and consistent structure is non-additive − The incompleteness both of quantum mechanics and arithmetic? – Choice, number, and probability − Ψ-function in a generalized notation – The sense of Einstein’s “common covariance” – “Princeton” also for gauge theories – More about “dualistic pythagoreanism” – Quantity and property – Projection operator as statement (à la von Neumann)− Simultaneous undecidability – Does the notion of physical quantity imply the invariance of time moments? – Commuting and non-commuting operators – Perfecting the notion for simultaneous immeasurability – Quantum mechanics in Procrustean bed – The world is also a mathematical structure for its essence
Quantum Measure from a Philosophical ViewpointVasil Penchev
1. The paper discusses quantum measure from a philosophical viewpoint. Quantum measure is a proposed three-dimensional measure that can measure both continuous and discrete phenomena uniformly.
2. Quantum measure is proposed as a more universal measure than Lebesgue or Borel measures. It is constructed to be invariant to the axiom of choice and resolves issues like how probability can become physical quantity.
3. The relationship between quantum mechanics and general relativity can be understood through this framework. They are complementary aspects of the quantum state of a system like the universe. However, general relativity is built assuming the axiom of choice while quantum mechanics is invariant to it, posing challenges for a unified theory of quantum gravity.
Quantum information as the information of infinite seriesVasil Penchev
The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The qubit, can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question.
Gödel’s completeness (1930) nd incompleteness (1931) theorems: A new reading ...Vasil Penchev
(T2) Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of inf(T1) Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation
Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity
That is nothing else than right a new reading at issue and comparative interpretation of Gödel’s papers meant here
inity
The most utilized example of those generalizations is the separable complex Hilbert space: it is able to equate the possibility of pure existence to the probability of statistical ensemble
(T3) Any generalization of Peano arithmetic consistent to infinity, e.g. the separable complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself
Action as the common measure of randomness and free willVasil Penchev
One needs a common measure of randomness and free well:
Both cause results, which are comparable
Given any fact without being known its origin, the question whether its origin is natural (i.e. random, occasional) or freewill action (intentional, constructive) arises
The result in both cases is one and the same as well as action and even the physical quantity of action
However the natural process being just random would require much more time in comparison with any intentional action for one and the same result
History as the ontology of time requires to be understood what time is
Time is the transformation of future into the past by the choices in the present
History should be grounded on that understanding of historical time, which would include the present and future rather than only the past
The thesis is: Einstein, Podolsky and Rosen’s argument (1935, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? ) is another interpretation of the famous Gödel incompleteness argument (1931, Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I ) in terms of quantum mechanics
Representation & Realityby Language (How to make a home quantum computer) Vasil Penchev
Reality as if is doubled in relation to language
We will model this doubling by two Turing machines (i.e. by usual computers) in a kind of “dialog”: the one for reality, the other for its image in language
The two ones have to reach the state of equilibrium to each other
At last, one can demonstrate that the pair of them is equivalent to a quantum computer
One can construct a model of two independent Turing machines allowing of a series of relevant interpretations:
Language
Quantum computer
Representation and metaphor
Reality and ontology
In turn that model is based on the concepts of choice and information
Математизирането на историята: число и битиеVasil Penchev
Книгата е първа част от по-широк замисъл − „Числа“. Така е назована на гръцки и от него на всички езици четвърта глава от Библията, посветена на преброяването на изра-илтяните и похода в пустинята. Стремежът е се да представи едно осъвременено пита-горейство. Предвиждат се още две части:
Синтактично-семантично интерпретиране на вълновата функция. Число и знак
Математизирането на историята. Число и история
Книгата е предназначена за научни работници в областта на философията, историята и математиката,
This document discusses quantum computers from a mathematical perspective by comparing them to Turing machines. It proposes that a quantum computer can be modeled as a Turing machine with an infinite tape of "qubits" rather than bits. This raises philosophical questions about the relationship between mathematical models and reality when dealing with infinity. The document also explores how concepts like information, choice, and measurement are understood differently in quantum as opposed to classical computation.
Quantum Mechanics as a Measure Theory: The Theory of Quantum MeasureVasil Penchev
This document discusses representing quantum mechanics as a measure theory, with the key points being:
1. Quantum measure can unify the measurement of discrete and continuous quantities by treating them as "much" and "many".
2. The unit of quantum measure is the qubit, allowing it to jointly measure probability, quantity, order, and disorder.
3. All physical processes can be interpreted as computations of a quantum computer, with the universal substance being quantum information.
4. Quantum measure can provide a nonlocal explanation for the Aharonov-Bohm effect by linking it to the electromagnetic nature of space-time itself.
The question is:
•
How should skepticism refer to itself?
•
The classical example might be the doubt of Descartes, which led him to the necessary obviousness of who doubts
•
The formal logical structure is the same as the “antinomy of the Liar”
•
That new interpretation of it can be called “antinomy of the Skeptic”
The document discusses interpretations of quantum mechanics including the Copenhagen interpretation, many-worlds interpretation, and transactional interpretation. It summarizes four quantum paradoxes around wave-particle duality, quantum measurement, and non-locality. It then provides more details on the transactional interpretation, explaining how it uses advanced and retarded waves to describe quantum events and resolve the paradoxes without needing observers or wavefunction collapse. Finally, it discusses how interpretations cannot be experimentally tested but notes one potential exception with a new experiment.
The document discusses quantum mechanics and three interpretations of its formalism: the Copenhagen interpretation, the many-worlds interpretation, and the transactional interpretation. It describes four quantum paradoxes around non-locality, wave-particle duality, and wave function collapse. Each interpretation aims to resolve these paradoxes while linking the mathematical formalism to physical phenomena.
"Possible Worlds and Substances“ by Vladislav TerekhovichVasil Penchev
The document discusses unifying the categories of possible worlds and substances through the concept of possible histories. It suggests describing the set of possible histories metaphysically as the gradual development of possible substances into distinguishable possible worlds.
The comment aims to mention related philosophical and scientific ideas, including unifying Gibbs and Boltzmann thermodynamics, describing discrete and continuous motions in quantum mechanics, and interpreting quantum mechanics' mathematical formalism through various metaphysical lenses.
It argues these ideas are members of the same "family" connected by a tendency toward mathematics. This could realize a form of Pythagoreanism identifying mathematics and reality, resolving problems in the conclusion of the original paper by identifying possibility as a universal probability substance underlying
Vasil Penchev. Gravity as entanglement, and entanglement as gravityVasil Penchev
1. The document discusses interpreting gravity as entanglement by investigating the conditions under which general relativity and quantum mechanics can be mapped to each other mathematically.
2. It outlines a strategy to interpret entanglement as inertial mass and gravitational mass, and to view gravity as another interpretation of any quantum mechanical or mechanical movement.
3. This would allow gravity to be incorporated into the standard model by generalizing the concept of quantum field to include entanglement, represented by a cyclic Yin-Yang mathematical structure.
The document discusses quantum mechanics and three interpretations of it: the Copenhagen interpretation, the Many-Worlds interpretation, and the Transactional interpretation. It describes some key aspects of each interpretation, such as how they explain wave function collapse, the role of observers, and how they address paradoxes in quantum mechanics like non-locality. The Copenhagen interpretation relies on observer knowledge, the Many-Worlds interpretation proposes the existence of parallel worlds, and the Transactional interpretation describes quantum events as transactions involving advanced and retarded waves.
It gives me great comfort to visualize this universe as the surface of an ever expanding four-dimensional sphere originating from a distant, but finite, past and growing indefinitely for ever. In this idealized model it easy to calculate the age of the universe by observing the velocity of the receding stars and also to make several other interesting conclusions. For more details, continue reading the presentation.
- In 1976, Stephen Hawking argued that black holes destroy information, requiring a modification of quantum mechanics principles. In 2004, he changed his mind.
- Maldacena's 1997 discovery of AdS/CFT duality suggested that a black hole is dual to an ordinary thermal system described by quantum mechanics, where information is preserved. However, questions remain about how spacetime emerges in AdS/CFT and how holography works in other spacetimes.
- A 2013 paper proposed that the postulates of black hole complementarity - purity, no drama at the horizon, effective field theory validity outside the horizon - cannot all be true, suggesting a "firewall" of high-energy particles may form at the black
Gravity as entanglement, and entanglement as gravityVasil Penchev
1) The document discusses the relationship between gravity and quantum entanglement, exploring the possibility that they are equivalent or closely connected concepts.
2) It outlines an approach to interpret gravity in terms of a generalized quantum field theory that includes entanglement, which could explain why gravity cannot be quantized.
3) The key idea is that entanglement expressed "outside" of space-time points looks like gravity "inside", and vice versa, with gravity representing a smooth constraint on the quantum behavior of entities imposed by all others.
This document summarizes three applications of commutative algebra to string theory. The first two applications involve interpreting certain products in topological field theory as Ext computations for sheaves on a Calabi-Yau manifold or in terms of matrix factorizations, which can be analyzed using computer algebra tools. The third application relates monodromy in string theory to solutions of differential equations, showing how monodromy can be described in terms of a computed ring.
Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding M...Premier Publishers
This document communicates some of the main results obtained from a theoretical work which performs a type of Wick’s rotation, where Lorentz’s group is connected in the resulting Euclidean metric, and as a consequence models the particles with rest mass as photons in a compacted additional dimension (for a photon of the ordinary 3-dimensional space, they do not go through the 4-dimension due to null angle in this dimension). Among its reported results are new explanations, much more elegant than the current ones, of the material waves of De Broglie, the uncertainty principle, the dilation of the proper time, the Higgs field, the existence of the antiparticles and specifically of the electron-positron annihilation, among others. It also leaves open the possibility of unifying at least three of the fundamental forces and the different types of particles under a single model of photon and compact dimension. Additionally, two experimental results are proposed that can only currently be explained by this theory.
Albert Einstein (2) Relativity Special And General TheoryKukuasu
This document provides instructions for classifying ebooks based on their file format and subject matter. It specifies that:
1) Ebooks should be in Adobe PDF or Tomeraider format, with txt files not considered ebooks.
2) The file name should include the classification in parenthesis - (ebook - File Format - Subject Matter).
3) The subject matter classification should be one of: Biography, Children, Fiction, Food, Games, Government, Health, Internet, Martial-Arts, Mathematics, Other, Programming, Reference, Religious, Science, Sci-Fi, Sex, or Software.
This standardization of ebook file names helps groups like Fink Crew
Quantum Geometry: A reunion of math and physicsRafa Spoladore
Caltech's professor Anton Kapustin "describes the relationship between mathematics and physics, mathematicians and physicists, and so on. He focuses on the noncommutative character of algebras of observables in quantum mechanics." via http://motls.blogspot.com.br/2014/11/anton-kapustin-quantum-geometry-reunion.html
Matter as Information. Quantum Information as MatterVasil Penchev
This document discusses interpreting matter and mass in physics as a form of quantum information. It argues that the concept of mass can be seen as a quantity of quantum information, with energy and matter interpreted as amounts of quantum information involved in infinite collections. Seeing mass and energy as quantum information helps unify the concepts of concrete and abstract objects by generalizing information from finite to infinite sets. This allows information to be viewed as a universal substance that subsumes the notions of mass and energy.
String theory is a theoretical framework that models particles as vibrating strings instead of point-like objects. It seeks to unite quantum mechanics and general relativity by incorporating gravity at small scales. In string theory, strings exist in 10 or 11 dimensions and vibrate in different ways. Their vibrational patterns determine properties like mass and charge, allowing strings to represent all known particles. String theory remains unproven, but efforts using machine learning may help explore its vast theoretical landscape.
This essay is a compilation of ideas, opinions, and conjectures from two previous essays, "Is Science Solving the Reality Riddle," and "Order, Chaos, and the End of Reductionism," and was expanded to include subsequent essays. It is very much a work in progress and has been repeatedly amended when necessary. The author concludes that current scientific theories are incomplete and limit our understanding of nature in a fundamental way, the current description of how the universe eveolved is wrong, and a new evolutionary paradigm is presented that explains both the physical and mental evolutionary processes.
Based on recent work on quantum gravity and the holographic principle I argue that, instead of thinking of the universe as a 'bubble out of nothing', we should think of space, time, and gravity as emerging 'out of information'.
Similar to Vasil Penchev. Continuity and Continuum in Nonstandard Universum (20)
The generalization of the Periodic table. The "Periodic table" of "dark matter"Vasil Penchev
The thesis is: the “periodic table” of “dark matter” is equivalent to the standard periodic table of the visible matter being entangled. Thus, it is to consist of all possible entangled states of the atoms of chemical elements as quantum systems. In other words, an atom of any chemical element and as a quantum system, i.e. as a wave function, should be represented as a non-orthogonal in general (i.e. entangled) subspace of the separable complex Hilbert space relevant to the system to which the atom at issue is related as a true part of it. The paper follows previous publications of mine stating that “dark matter” and “dark energy” are projections of arbitrarily entangled states on the cognitive “screen” of Einstein’s “Mach’s principle” in general relativity postulating that gravitational field can be generated only by mass or energy.
Modal History versus Counterfactual History: History as IntentionVasil Penchev
The distinction of whether real or counterfactual history makes sense only post factum. However, modal history is to be defined only as ones’ intention and thus, ex-ante. Modal history is probable history, and its probability is subjective. One needs phenomenological “epoché” in relation to its reality (respectively, counterfactuality). Thus, modal history describes historical “phenomena” in Husserl’s sense and would need a specific application of phenomenological reduction, which can be called historical reduction. Modal history doubles history just as the recorded history of historiography does it. That doubling is a necessary condition of historical objectivity including one’s subjectivity: whether actors’, ex-anteor historians’ post factum. The objectivity doubled by ones’ subjectivity constitute “hermeneutical circle”.
Both classical and quantum information [autosaved]Vasil Penchev
Information can be considered a the most fundamental, philosophical, physical and mathematical concept originating from the totality by means of physical and mathematical transcendentalism (the counterpart of philosophical transcendentalism). Classical and quantum information. particularly by their units, bit and qubit, correspond and unify the finite and infinite:
As classical information is relevant to finite series and sets, as quantum information, to infinite ones. The separable complex Hilbert space of quantum mechanics can be represented equivalently as “qubit space”) as quantum information and doubled dually or “complimentary” by Hilbert arithmetic (classical information).
A CLASS OF EXEMPLES DEMONSTRATING THAT “푃푃≠푁푁푁 ” IN THE “P VS NP” PROBLEMVasil Penchev
The CMI Millennium “P vs NP Problem” can be resolved e.g. if one shows at least one counterexample to the “P=NP” conjecture. A certain class of problems being such counterexamples will be formulated. This implies the rejection of the hypothesis “P=NP” for any conditions satisfying the formulation of the problem. Thus, the solution “P≠NP” of the problem in general is proved. The class of counterexamples can be interpreted as any quantum superposition of any finite set of quantum states. The Kochen-Specker theorem is involved. Any fundamentally random choice among a finite set of alternatives belong to “NP’ but not to “P”. The conjecture that the set complement of “P” to “NP” can be described by that kind of choice exhaustively is formulated.
FERMAT’S LAST THEOREM PROVED BY INDUCTION (accompanied by a philosophical com...Vasil Penchev
A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n=3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite descent is linked to induction starting from n=3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n=4, one can suggest that the proof for n≥4 was accessible to him.
An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995), and would justify Fermat’s claim (1637) for its proof. The inspiration for a simple proof would contradict to Descartes’s dualism for appealing to merge “mind” and “body”, “words” and “things”, “terms” and “propositions”, all orders of logic. A counterfactual course of history of mathematics and philosophy may be admitted. The bifurcation happened in Descartes and Fermat’s age. FLT is exceptionally difficult to be proved in our real branch rather than in the counterfactual one.
The space-time interpretation of Poincare’s conjecture proved by G. Perelman Vasil Penchev
This document discusses the generalization of Poincaré's conjecture to higher dimensions and its interpretation in terms of special relativity. It proposes that Poincaré's conjecture can be generalized to state that any 4-dimensional ball is topologically equivalent to 3D Euclidean space. This generalization has a physical interpretation in which our 3D space can be viewed as a "4-ball" closed in a fourth dimension. The document also outlines ideas for how one might prove this generalization by "unfolding" the problem into topological equivalences between Euclidean spaces.
FROM THE PRINCIPLE OF LEAST ACTION TO THE CONSERVATION OF QUANTUM INFORMATION...Vasil Penchev
In fact, the first law of conservation (that of mass) was found in chemistry and generalized to the conservation of energy in physics by means of Einstein’s famous “E=mc2”. Energy conservation is implied by the principle of least action from a variational viewpoint as in Emmy Noether’s theorems (1918): any chemical change in a conservative (i.e. “closed”) system can be accomplished only in the way conserving its total energy. Bohr’s innovation to found Mendeleev’s periodic table by quantum mechanics implies a certain generalization referring to
the quantum leaps as if accomplished in all possible trajectories (according to Feynman’s interpretation) and therefore generalizing the principle of least action and needing a certain generalization of energy conservation as to any quantum change.The transition from the first to the second theorem of Emmy Noether represents well the necessary generalization: its chemical meaning is the ge eralization of any chemical reaction to be accomplished as if any possible course of time rather than in the standard evenly running time (and equivalent to energy conservation according to the first theorem). The problem: If any quantum change is accomplished in al possible “variations (i.e. “violations) of energy conservation” (by different probabilities),
what (if any) is conserved? An answer: quantum information is what is conserved. Indeed, it can be particularly defined as the counterpart (e.g. in the sense of Emmy Noether’s theorems) to the physical quantity of action (e.g. as energy is the counterpart of time in them). It is valid in any course of time rather than in the evenly running one. That generalization implies a generalization of the periodic table including any continuous and smooth transformation between two chemical elements.
From the principle of least action to the conservation of quantum information...Vasil Penchev
In fact, the first law of conservation (that of mass) was found in chemistry and generalized to the conservation of energy in physics by means of Einstein’s famous “E=mc2”. Energy conservation is implied by the principle of least action from a variational viewpoint as in Emmy Noether’s theorems (1918):any chemical change in a conservative (i.e. “closed”) system can be accomplished only in the way conserving its total energy. Bohr’s innovation to found Mendeleev’s periodic table by quantum mechanics implies a certain generalization referring to the quantum leaps as if accomplished in all possible trajectories (e.g. according to Feynman’s viewpoint) and therefore generalizing the principle of least action and needing a certain generalization of energy conservation as to any quantum change.
The transition from the first to the second theorem of Emmy Noether represents well the necessary generalization: its chemical meaning is the generalization of any chemical reaction to be accomplished as if any possible course of time rather than in the standard evenly running time (and equivalent to energy conservation according to the first theorem).
The problem: If any quantum change is accomplished in all possible “variations (i.e. “violations) of energy conservation” (by different probabilities), what (if any) is conserved?
An answer: quantum information is what is conserved. Indeed it can be particularly defined as the counterpart (e.g. in the sense of Emmy Noether’s theorems) to the physical quantity of action (e.g. as energy is the counterpart of time in them). It is valid in any course of time rather than in the evenly running one. (An illustration: if observers in arbitrarily accelerated reference frames exchange light signals about the course of a single chemical reaction observed by all of them, the universal viewpoint shareаble by all is that of quantum information).
That generalization implies a generalization of the periodic table including any continuous and smooth transformation between two chemical elements necessary conserving quantum information rather than energy: thus it can be called “alchemical periodic table”.
Poincaré’s conjecture proved by G. Perelman by the isomorphism of Minkowski s...Vasil Penchev
- The document discusses the relationship between separable complex Hilbert spaces (H) and sets of ordinals (H) and how they should not be equated if natural numbers are identified as finite.
- It presents two interpretations of H: as vectors in n-dimensional complex space or as squarely integrable functions, and discusses how the latter adds unitarity from energy conservation.
- It argues that Η rather than H should be used when not involving energy conservation, and discusses how the relation between H and HH generates spheres representing areas and can be interpreted physically in terms of energy and force.
Why anything rather than nothing? The answer of quantum mechnaicsVasil Penchev
Many researchers determine the question “Why anything
rather than nothing?” to be the most ancient and fundamental philosophical problem. It is closely related to the idea of Creation shared by religion, science, and philosophy, for example in the shape of the “Big Bang”, the doctrine of first cause or causa sui, the Creation in six days in the Bible, etc. Thus, the solution of quantum mechanics, being scientific in essence, can also be interpreted philosophically, and even religiously. This paper will only discuss the philosophical interpretation. The essence of the answer of quantum mechanics is: 1.) Creation is necessary in a rigorously mathematical sense. Thus, it does not need any hoice, free will, subject, God, etc. to appear. The world exists by virtue of mathematical necessity, e.g. as any mathematical truth such as 2+2=4; and 2.) Being is less than nothing rather than ore than nothing. Thus creation is not an increase of nothing, but the decrease of nothing: it is a deficiency in relation to nothing. Time and its “arrow” form the road from that diminishment or incompleteness to nothing.
The Square of Opposition & The Concept of Infinity: The shared information s...Vasil Penchev
The power of the square of opposition has been proved during millennia, It supplies logic by the ontological language of infinity for describing anything...
6th WORLD CONGRESS ON THE SQUARE OF OPPOSITION
http://www.square-of-opposition.org/square2018.html
Mamardashvili, an Observer of the Totality. About “Symbol and Consciousness”,...Vasil Penchev
The paper discusses a few tensions “crucifying” the works and even personality of the great Georgian philosopher Merab Mamardashvili: East and West; human being and thought, symbol and consciousness, infinity and finiteness, similarity and differences. The observer can be involved as the correlative counterpart of the totality: An observer opposed to the totality externalizes an internal part outside. Thus the phenomena of an observer and the totality turn out to converge to each other or to be one and the same. In other words, the phenomenon of an observer includes the singularity of the solipsistic Self, which (or “who”) is the same as that of the totality. Furthermore, observation can be thought as that primary and initial action underlain by the phenomenon of an observer. That action of observation consists in the externalization of the solipsistic Self outside as some external reality. It is both a zero action and the singularity of the phenomenon of action. The main conclusions are: Mamardashvili’s philosophy can be thought both as the suffering effort to be a human being again and again as well as the philosophical reflection on the genesis of thought from itself by the same effort. Thus it can be recognized as a powerful tension between signs anа symbol, between conscious structures and consciousness, between the syncretism of the East and the discursiveness of the West crucifying spiritually Georgia
Completeness: From henkin's Proposition to Quantum ComputerVasil Penchev
This document discusses how Leon Henkin's proposition relates to concepts in logic, set theory, information theory, and quantum mechanics. It argues that Henkin's proposition, which states the provability of a statement within a formal system, is equivalent to an internal and consistent position regarding infinity. The document then explores how this connects to Martin Lob's theorem, the Einstein-Podolsky-Rosen paradox in quantum mechanics, theorems about the absence of hidden variables, entanglement, quantum information, and ultimately quantum computers.
Why anything rather than nothing? The answer of quantum mechanicsVasil Penchev
This document discusses the philosophical question of why there is something rather than nothing from the perspective of quantum mechanics. It argues that quantum mechanics provides a solution where creation is permanent and due to the irreversibility of time. The creation in quantum mechanics represents a necessary loss of information as alternatives are rejected in the course of time, rather than being due to some external cause like God's will. This permanent creation process makes the universe mathematically necessary rather than requiring an initial singular event like the Big Bang.
The outlined approach allows a common philosophical viewpoint to the physical world, language and some mathematical structures therefore calling for the universe to be understood as a joint physical, linguistic and mathematical universum, in which physical motion and metaphor are one and the same rather than only similar in a sense.
Hilbert Space and pseudo-Riemannian Space: The Common Base of Quantum Informa...Vasil Penchev
Hilbert space underlying quantum mechanics and pseudo-Riemannian space underlying general relativity share a common base of quantum information. Hilbert space can be interpreted as the free variable of quantum information, and any point in it, being equivalent to a wave function (and thus, to a state of a quantum system), as a value of that variable of quantum information. In turn, pseudo-Riemannian space can be interpreted as the interaction of two or more quantities of quantum information and thus, as two or more entangled quantum systems. Consequently, one can distinguish local physical interactions describable by a single Hilbert space (or by any factorizable tensor product of such ones) and non-local physical interactions describable only by means by that Hilbert space, which cannot be factorized as any tensor product of the Hilbert spaces, by means of which one can describe the interacting quantum subsystems separately. Any interaction, which can be exhaustedly described in a single Hilbert space, such as the weak, strong, and electromagnetic one, is local in terms of quantum information. Any interaction, which cannot be described thus, is nonlocal in terms of quantum information. Any interaction, which is exhaustedly describable by pseudo-Riemannian space, such as gravity, is nonlocal in this sense. Consequently all known physical interaction can be described by a single geometrical base interpreting it in terms of quantum information.
This document discusses using Richard Feynman's interpretation of quantum mechanics as a way to formally summarize different explanations of quantum mechanics given to hypothetical children. It proposes that each child's understanding could be seen as one "pathway" or explanation, with the total set of explanations forming a distribution. The document then suggests that quantum mechanics itself could provide a meta-explanation that encompasses all the children's perspectives by describing phenomena probabilistically rather than deterministically. Finally, it gives some examples of how this approach could allow defining and experimentally studying the concept of God through quantum mechanics.
This document discusses whether artificial intelligence can have a soul from both scientific and religious perspectives. It begins by acknowledging that "soul" is a religious concept while AI is a scientific one. The document then examines how Christianity views creativity as a criterion for having a soul. It proposes formal scientific definitions of creativity involving learning rates and probabilities. An example is given comparing a master's creativity to an apprentice's. The document argues science can describe God's infinite creativity and human's finite creativity uniformly. It analyzes whether criteria for creativity can apply to AI like a Turing machine. Hypothetical examples involving infinite algorithms and self-learning machines are discussed.
Analogia entis as analogy universalized and formalized rigorously and mathema...Vasil Penchev
THE SECOND WORLD CONGRESS ON ANALOGY, POZNAŃ, MAY 24-26, 2017
(The Venue: Sala Lubrańskiego (Lubrański’s Hall at the Collegium Minus), Adam Mickiewicz University, Address: ul. Wieniawskiego 1) The presentation: 24 May, 15:30
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Vasil Penchev. Continuity and Continuum in Nonstandard Universum
1. Continuity and Continuum in
Nonstandard Universum
Vasil Penchev
Institute of Philosophical Research
Bulgarian Academy of Science
E-mail: vasildinev@gmail.com
Publications blog:
http://www.esnips.com/web/vasilpenchevsnews
2. Content
1. Motivation
s:
2. Infinity and the axiom of choice
3. Nonstandard universum
4. Continuity and continuum
5. Nonstandard continuity
between two infinitely close
standard points
6. A new axiom: of chance
7. Two kinds interpretation of
3. This file is only Part 1 of the
entire presentation and
includes:
1. Motivation
2. Infinity and the axiom of choice
3. Nonstandard universum
4. : 1. Motivation :
My problem was:
Given: Two sequences:
: 1, 2, 3, 4, ….a-3, a-2, a-1, a
: a, a-1, a-2, a-3, …, 4, 3, 2, 1
Where a is the power of
countable set
The problem:
Do the two sequences and
5. : 1. Motivation :
At last, my resolution proved
out:
That the two sequences:
: 1, 2, 3, 4, ….a-3, a-2, a-1, a
: a, a-1, a-2, a-3, …, 4, 3, 2, 1
coincide or not, is a new axiom (or
two different versions of the
choice axiom): the axiom of
6. : 1. Motivation :
For example, let us be given two
Hilbert spaces:
: eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat
: eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eit
An analogical problem is:
Are those two Hilbert spaces the
same or not?
can be got by Minkowski space
after Legendre-like
7. : 1. Motivation :
So that, if:
: eit, ei2t, ei3t, ei4t, … ei(a-1)t, eiat
: eiat , ei(a-1)t, … ei4t, ei3t, ei2t, eit
are the same, then Hilbert space
is equivalent of the set of all
the continuous world lines in
spacetime
(see also Penrose’s twistors)
That is the real problem, from
8. : 1. Motivation :
About that real problem, from
which I had started, my
conclusion was:
There are two different versions
about the transition between the
micro-object Hilbert space and
the apparatus spacetime in
dependence on accepting or
rejecting of ‚the chance
9. : 1. Motivation :
After that, I noticed that the
problem is very easily to be
interpreted by transition within
nonstandard universum between
two nonstandard neighborhoods
(ultrafilters) of two infinitely
near standard points or between
the standard subset and the
properly nonstandard subset of
10. : 1. Motivation :
And as a result, I decided that
only the
highly respected scientists from
the honorable and reverend
department ‚Logic‛ are that
appropriate public worthy and
deserving of being delivered
a report on that most intriguing
11. : 1. Motivation :
After that, the very God was so
benevolent so that He allowed
me to recognize marvelous
mathematical papers of a great
Frenchman, Alain
Connes, recently who has
preferred in favor of sunny
California to settle, and who, a
long time ago, had introduced
12. Content
1. Motivation
s:
2. INFINITY and the AXIOM OF CHOICE
3. Nonstandard universum
4. Continuity and continuum
5. Nonstandard continuity between
two infinitely close standard
points
6. A new axiom: of chance
7. Two kinds interpretation of
13. Infinity and the Axiom of
Choice
A few preliminary notes about
how the knowledge of infinity is
possible: The short answer is: as
that of God: in belief and by
analogy.The way of mathematics
to be achieved a little
knowledge of infinity transits
three stages: 1. From finite
perception to Axioms 2. Negation
14. Infinity and the Axiom of
Choice
The way of mathematics to
infinity:
1. From our finite experience and
perception to Axioms: The most
famous example is the
axiomatization of geometry
accomplished by Euclid in his
‚Elements‛
15. Infinity and the Axiom of
Choice
The way of mathematics to
infinity:
2. Negation of some axioms: the
most frequently cited instance is
the fifth Euclid postulate and its
replacing in Lobachevski
geometry by one of its negations.
Mathematics only starts from
16. Infinity and the Axiom of
Choice
The way of mathematics to
infinity:
3. Mathematics beyond finiteness:
We can postulate some
properties of infinite sets by
analogy of finite ones (e.g.
‘number of elements’ and
‘power’) However such transfer
17. Infinity and the Axiom of
Choice
A few inferences about the math
full-scale offensive amongst the
infinity:
1. Analogy: well-chosen
appropriate properties of finite
mathematical struc-tures are
transferred into infinite ones
2. Belief: the transferred
18. Infinity and the Axiom of
Choice
The most difficult problems of
the math offensive among
infinity:
1. Which transfers are allowed
by in-finity without producing
paradoxes?
2. Which properties are suitable
19. Infinity and the Axiom of
Choice
The Axiom of Choice (a
formulation):
If given a whatever set A
consisting of sets, we always can
choose an element from each
set, thereby constituting a new
set B (obviously of the same po-
wer as A). So its sense is: we
20. Infinity and the Axiom of
Choice
Some other formulations or
corollaries:
1. Any set can be well ordered
(any its subset has a least
element)
2. Zorn’s lema
3. Ultrafilter lema
4. Banach-Tarski paradox
21. Infinity and the Axiom of
Choice
Zorn’s lemma is equivalent to the
axiom of choice. Call a set A a
chain if for any two members B
and C, either B is a sub-set of C or C
is a subset of B. Now con-sider a
set D with the properties that for
every chain E that is a subset of
D, the union of E is a member of D.
The lem-ma states that D contains
a member that is maximal, i.e.
22. Infinity and the Axiom of
Choice
Ultrafilter lemma: A filter on
a set X is a collection of
nonempty subsets of X that is
closed under finite
intersection and under
superset. An ultrafilter is a
maximal filter. The
ultrafilter lemma states that
every filter on a set X is a
23. Infinity and the Axiom of
Choice
Banach–Tarski paradox which
says in effect that it is possible
to ‘carve up’ the 3-dimensional
solid unit ball into finitely many
pieces and, using only rotation
and translation, reassemble the
pieces into two balls each with
the same volume as the original.
The proof, like all proofs
24. Infinity and the Axiom of
Choice
First stated in 1924, the Banach-
Tarski paradox states that it is
possible to dissect a ball into
six pieces which can be
reassembled by rigid motions to
form two balls of the same size
as the original. The number of
pieces was subsequently reduced
to five by Robinson
25. Infinity and the Axiom of
Choice
Five pieces are minimal, although
four pieces are sufficient as long
as the single point at the center
is neglected. A generalization of
this theorem is that any two
bodies in that do not extend to
infinity and each containing a
ball of arbitrary size can be
dissected into each other (i.e.,
26. Infinity and the Axiom of
Choice
Banach-Tarski paradox is very
important for quantum
mechanics and information since
any qubit is isomorphic to a 3D
sphere. That’s why the paradox
requires for arbitrary qubits
(even entire Hilbert space) to be
able to be built by a single qubit
from its parts by translations
27. Infinity and the Axiom of
Choice
So that the Banach-Tarski
paradox implies the phenomenon
of entanglement in quantum
information as two qubits (or
two spheres) from one can be
considered as thoroughly
entangled. Two partly entangled
qubits could be reckoned as
sharing some subset of an initial
28. Infinity and the Axiom of
Choice
But the Banach-Tarski paradox is
a weaker statement than the
axiom of choice. It is valid only
about 3D sets. But I haven’t meet
any other additional condition.
Let us accept that the Banach-
Tarski paradox is equivalent to
the axiom of choice for 3D sets.
But entanglement as well 3D
29. Infinity and the Axiom of
Choice
But entanglement (= Banach-
Tarski paradox) as well 3D
space are physical facts, and
then consequently, they are
empirical confirmations in favor
of the axiom of choice. This
proves that the Banach-Tarski
paradox is just the most decisive
confirmation, and not at all, a
30. Infinity and the Axiom of
Choice
Besides, the axiom of choice
occurs in the proofs of: the Hahn-
Banach the-orem in functional
analysis, the theo-rem that
every vector space has a ba-
sis, Tychonoff's theorem in
topology stating that every
product of compact spaces is
compact, and the theorems in
abstract algebra that every ring
31. Infinity and the Axiom of
Choice
The Continuum Hypothesis:
The generalized continuum
hypothesis (GCH) is not only
independent of ZF, but also
independent of ZF plus the axiom
of choice (ZFC). However, ZF plus
GCH implies AC, making GCH a
strictly stronger claim than
AC, even though they are both
32. Infinity and the Axiom of
Choice
The Continuum Hypothesis:
The generalized continuum
hypothesis (GCH) is: 2Na = Na+1 . Since
it can be formulated without
AC, entanglement as an argument
in favor of AC is not expanded to
GCH. We may assume the negation
of GHC about cardinalities which
are not ‚alefs‛ together with AC
33. Infinity and the Axiom of
Choice
Negation of Continuum Hypothesis:
The negation of GHC about
cardinali-ties which are not
‚alefs‛ together with AC about
cardinalities which are alefs:
1. There are sets which can not be
well ordered. A physical
interpretation of theirs is as
physical objects out of (beyond)
34. Infinity and the Axiom of
Choice
Negation of Continuum Hypothesis:
But the physical sense of 1. and 2.:
1. The non-well-orderable sets
consist of well-ordered subsets
(at least, their elements as sets)
which are together in space-time.
2. Any well-ordered set (because
of Banach-Tarski paradox) can be
as a set of entangled objects in
35. Infinity and the Axiom of
Choice
Negation of Continuum Hypothesis:
So that the physical sense of 1.
and 2. is ultimately: The mapping
between the set of space-time
points and the set of physical
entities is a ‚many-many‛
correspondence: It can be
equivalently replaced by usual
mappings but however of a
36. Infinity and the Axiom of
Choice
Negation of Continuum Hypothesis:
Since the physical quantities have
interpreted by Hilbert
operators in quantum mechanics
and information
(correspondingly, by Hermitian
and non-Hermitian ones), then
that fact is an empirical
37. Infinity and the Axiom of
Choice
Negation of Continuum Hypothesis:
But as well known, ZF+GHC
implies AC. Since we have already
proved both NGHC and AC, the
only possibility remains also the
negation of ZF (NZF), namely the
negation the axiom of foundation
(AF): There is a special kind of
sets, which will call ‘insepa-
38. Infinity and the Axiom of
Choice
An important example of
inseparable set: When
postulating that if a set A is
given, then a set B always
exists, such one that A is the set
of all the subsets of B. An
instance: let A be a countable
set, then B is an inseparable
set, which we can call
39. Infinity and the Axiom of
Choice
The axiom of foundation: ‚Every
nonempty set is disjoint from
one of its elements.‚ It can also
be stated as "A set contains no
infinitely descending
(membership) sequence," or "A
set contains a (membership)
minimal element," i.e., there is an
40. Infinity and the Axiom of
Choice
The axiom of foundation
Mendelson (1958) proved that the
equivalence of these two
statements necessarily relies on
the axiom of choice. The dual
expression is called
º-induction, and is equivalent to
41. Infinity and the Axiom of
Choice
The axiom of foundation and its
negation: Since we have accepted
both the axiom of choice and the
negation of the axiom of
foundation, then we are to
confirm the negation of º-
induction, namely ‚There are sets
containing infinitely descending
(membership) sequence OR
42. Infinity and the Axiom of
Choice
The axiom of foundation and its
negation: So that we have three
kinds of inseparable set:
1.‚containing infinitely descending
(membership) sequence‛ 2.
‚without a (membership) minimal
element‚ 3. Both 1. and 2.
The alleged ‚axiom of chance‛
43. Infinity and the Axiom of
Choice
The alleged ‚axiom of chance‛
concerning only 1. claims that
there are as inseparable sets
‚containing infinitely descending
(membership) sequence‛ as such
ones ‚containing infinitely
ascending (membership)
sequence‛ and different from
44. Infinity and the Axiom of
Choice
The Law of the excluded middle:
The assumption of the axiom of
choice is also sufficient to derive
the law of the excluded middle
in some constructive systems
(where the law is not assumed).
45. Infinity and the Axiom of
Choice
A few (maybe redundant)
commentaries:
We always can:
1. Choose an element among the
elements of a set of an
arbitrary power
2. Choose a set among the
46. Infinity and the Axiom of
Choice
A (maybe rather useful)
commentary:
We always can:
3a. Repeat the choice choosing the
same element according to 1.
3b. Repeat the choice choosing the
same set according to 2.
47. Infinity and the Axiom of
Choice
The sense of the Axiom of Choice:
1. Choice among infinite elements
2. Choice among infinite sets
3. Repetition of the already
made choice among infinite
elements
4. Repetition of the already
48. Infinity and the Axiom of
Choice
The sense of the Axiom of Choice:
If all the 1-4 are fulfilled:
- choice is the same as among
finite as among infinite
elements or sets;
- the notion of information being
based on choice is the same as
49. Infinity and the Axiom of
Choice
At last, the award for your kind
patience: The linkages between
my motivation and the choice
axiom:
When accepting its negation, we
ought to recognize a special kind
of choice and of information in
relation of infinite entities:
50. Infinity and the Axiom of
Choice
So that the axiom of choice
should be divided into two parts:
The first part concerning
quantum choice claims that the
choice between infinite elements
or sets is always possible. The
second part concerning quantum
information claims that the
made already choice between
51. Infinity and the Axiom of
Choice
My exposition is devoted to the
nega-tion only of the ‚second
part‛ of the choice axiom. But
not more than a couple of words
about the sense for the first
part to be replaced or canceled:
When doing that, we accept a new
kind of entities: whole without
parts in prin-ciple, or in other
words, such kind of
52. Infinity and the Axiom of
Choice
Negating the choice axiom second
part is the suggested ‚axiom of
chance‛ properly speaking. Its
sense is: quantum information
exists, and it is different than
‚classical‛ one. The former
differs from the latter in five
basic properties as following:
copying, destroying, non-self-
interacting, energetic
53. Infinity and the Axiom of
Choice
Classical
Quantum
1. Copying, Yes No
2. Destroying, Yes No
3. Non-self-interacting, Yes No
4. Energetic medium, Yes No
5. Being in space-time Yes No
54. Infinity and the Axiom of
Choice
How does the ‚1. Copying‛
(Yes/No) descend from
(No/Yes)?
It is obviously: ‚Copying‛ means
that a set of choices is
repeated, and
consequently, it has been able to
55. Infinity and the Axiom of
Choice
If the case is: ‚1. Copying – No‛
from
- Yes,
then that case is the non-cloning
theorem in quantum information:
No qubit can be copied
56. Infinity and the Axiom of
Choice
How does the ‚2. Destroying‛
(Yes/No) descend from
(No/Yes)?
‚Destroying‛ is similar to
copying:
As if negative copying
57. Infinity and the Axiom of
Choice
How does the ‚3. Non-self-
interacting‛ (Yes/No) descend
from
(No/Yes)?
Self-interacting means
non-repeating by itself
58. Infinity and the Axiom of
Choice
How does the ‚4. Energetic
medium‛ (Yes/No) descend
from
(No/Yes)?
Energetic medium means for
repeating to be turned into
substance, or in other words, to
59. Infinity and the Axiom of
Choice
How does the ‚5. Being in space-
time‛ (Yes/No) descend from
(No/Yes)?
‘Being of a set in space-time’
means that the set is well-
ordered which fol-lows from
the axiom of choice. ‘No axiom of
chance’ means that the well-
60. Content
1. Motivation
s:
2. Infinity and the axiom of choice
3. NONSTANDARD UNIVERSUM
4. Continuity and continuum
5. Nonstandard continuity between
two infinitely close standard
points
6. A new axiom: of chance
7. Two kinds interpretation of
62. Nonstandard universum
Abraham Robinson
(October 6, 1918
His Book (1966) – April 11, 1974)
63. Nonstandard universum
‚It is shown in this
book that Leibniz
ideas can be fully
vindicated and that
they lead to a
novel and fruitful
approach to
classical Analysis
His Book (1966) and many other
branches of
64. Nonstandard universum
‚…G.W.Leibniz argued that
the theory of infinitesimals
implies the introduction of
ideal numbers which might
be infinitely small or
infinitely large compared
with the real numbers but
which were to possess the
65. Nonstandard universum
The original approach of A.
Robinson:
1. Construction of a nonstandard
model of R (the real continuum):
Nonstan-dard model (Skolem
1934): Let A be the set of all the
true statements about R, then: =
A(c>0, c>0`, c>0``…): Any finite
subset of holds for R. After
66. Nonstandard universum
2. The finiteness principle: If
any fi-nite subset of a (infinite)
set posses-ses a model, then
the set possesses a model too.
The model of is not isomorphic
to R & A and it is a nonstandard
universum over R & A. Its sense is
as follow: there is a
nonstandard neighborhood x
67. Nonstandard universum
The properties of nonstandard
neighborhood x about any
standard point x of R: 1) The
‚length‛ of x in R or of any its
measurable subset is 0. 2) Any x
in R is isomorphic to (R & A)
itself. Our main problem is about
continuity and continuum of two
neighborhoods x and y
between two neighbor well
68. Nonstandard universum
Indeed, the word of G.W.Leibniz
‚that the theory of infinitesimals
implies the introduction of ideal
numbers which might be
infinitely small or infinitely
large compared with the real
numbers but which were to
possess the same properties as
the latter‛ (Robinson, p. 2) are
69. Nonstandard universum
Another possible approach was
developed by was developed in
the mid-1970s by the
mathematician Edward Nelson.
Nelson introduced an entirely
axiomatic formulation of non-
standard analysis that he called
Internal Set Theory or IST. IST is
an extension of Zermelo-
70. Nonstandard universum
In IST alongside the basic binary
membership relation , it
introduces a new unary predicate
standard which can be applied to
elements of the mathematical
universe together with three
axioms for reasoning with this
new predicate (again IST): the
axioms of
71. Nonstandard universum
Idealization:
For every classical relation R, and
for arbit-rary values for all other
free variables, we have that if for
each standard, finite set F, there
exists a g such that R(g, f ) holds for
all f in F, then there is a particular G
such that for any standard f we have
R (G, f ), and conversely, if there
exists G such that for any standard
f, we have R(G, f ), then for each
72. Nonstandard universum
Standardisation
If A is a standard set and P any
property, classical or
otherwise, then there is a
unique, standard subset B of A
whose standard elements are
precisely the standard elements
of A satisfying P (but the
behaviour of B's nonstandard
73. Nonstandard universum
Transfer
If all the parameters
A, B, C, ..., W
of a classical formula F have
standard values then
F( x, A, B,..., W )
holds for all x's as soon as
it holds for all standard xs.
74. Nonstandard universum
The sense of the unary predicate
standard:
If any formula holds for any finite
standard
set of standard elements, it holds
for all the universum. So that
standard elements are only those
which establish, set the
standards, with which all the
elements must be in conformity: In
75. Nonstandard universum
So that the suggested by Nelson IST is
a constructivist version of
nonstandard analysis. If ZFC is
consistent, then ZFC + IST is consistent.
In fact, a stronger statement can be
made: ZFC + IST is a conservative
extension of ZFC: any classical
formula (correct or incorrect!) that
can be proven within internal set
theory can be proven in the
76. Nonstandard universum
The basic idea of both the version
of nonstandard analysis (as
Roninson’s as Nelson’s) is
repetition of all the real
continuum R at, or better, within
any its point as nonstandard
neighborhoods about any of
them. The consistency of that
repetition is achieved by the
77. Nonstandard universum
That collapse and repetition of
all infinity into any its point is
accomp-lished by the notion of
ultrafilter in nonstandard
analysis. Ultrafilter is way to be
transferred and thereby
repeated the topological
properties of all the real
continuum into any its point, and
after that, all the properties of
78. Nonstandard universum
What is ‘ultrafilter’?
Let S be a nonempty set, then an
ultrafilter on S is a nonempty
collection F of subsets of S having
the following properties:
1. F.
2. If A, B F, then A, B F .
3. If A,B F and ABS, then A,B F
4. For any subset A of S, either A F
79. Nonstandard universum
Ultrafilter lemma: A filter on
a set X is a collection of
nonempty subsets of X that is
closed under finite intersection
and under superset. An
ultrafilter is a maximal filter.
The ultrafilter lemma states
that every filter on a set X is a
subset of some ultrafilter on X
80. Nonstandard universum
A philosophical reflection: Let us
remember the Banach-Tarski
paradox: entire Hilbert space can be
delivered only by repetition ad
infinitum of a single qubit (since it is
isomorphic to 3D sphere)as well the
paradox follows from the axiom of
choice. However nonstandard
analysis carries out the same idea as the
Banach-Tarski paradox about 1D sphere, i.e.
a point: all the nonstandard universum can
81. Nonstandard universum
The philosophical reflection
continues: That’s why nonstandard
analysis is a good tool for quantum
mechanics: Nonstandard universum
(NU) possesses as if fractal structure
just as Hilbert space. It allows all
quantum objects to be described as
internal sets absolutely similar to
macro-objects being described as
external or standard sets. The best
advantage is that NU can describe the
82. Nonstandard universum
Something still a little more: If
Hilbert spa-ce is isomorphic to a
well ordered sequence of 3D
spheres delivered by the axiom of
choice via the Banach-Tarski
paradox, then 1. It is at least
comparable unless even iso-morphic
to Minkowski space; 2. It is getting
generalized into nonstandard
universum as to arbitrary number
dimensions, and even as to fractional
83. Nonstandard universum
And at last: The generalized so
Hilbert space as nonstandard
universum is delivered again by the
axiom of choice but this time via
Zorn’s lemma (an equivalent to the
axiom of choice) via ultrafilter
lemma (a weaker statement than the
axiom of choice). Nonstandard
universum admits to be in its turn
generalized as in the gauge
theories, when internal and
84. Nonstandard universum
Thus we have already pioneered to
Alain Connes’ introducing of
infinitesimals as compact Hilbert
operators unlike the rest Hilbert
operators representing transfor-
mations of standard sets. He has
suggested the following
‚dictionary‛:
Complex variable Hilbert
85. Nonstandard universum
The sense of compact operator: if it
is ap-plied to nonstandard
universum, it trans-forms a
nonstandard neighborhood into a
nonstandard neighborhood, so that
it keeps division between standard
and nonstandard elements. If the
nonstandard universum is built on
Hilbert space instead of on real
continuum, then Connes defined
infinite-simals on the Cartesian
86. Nonstandard universum
I would like to display that Connes’
infinitesimals possesses an
exceptionally important property:
they are infinitesimals both in
Hilbert and in Minkowski space: so
that they describe very well
transformations of Minkowski space
into Hilbert space and vice versa:
Math speaking, Minkowski operator
is compact if and only if it is
compact Hilbert operator. You might
87. Nonstandard universum
Minkowski operator is compact if
and only if it is compact Hilbert
operator. Before a sketch of
proof, its sense and motivation: If we
describe the transformations of
Minkow-ski space into Hilbert space
and vice versa, we will be able to
speak of the transition between the
apparatus and the microobject and
vice versa as well of the transition
bet-ween the coherent and
88. Nonstandard universum
Before a sketch of proof, its sense
and motivation: Our strategic
purpose is to be built a
united, common language for us to
be able to speak both of the
apparatus and of the microobject as
well, and the most impor-tant, of
the transition and its converse bet-
ween them. The creating of such a
language requires a different set-
theory foundation including: 1. The
89. Nonstandard universum
Before a sketch of proof, its sense
and motivation: The axiom of
foundation is available in quantum
mechanics by the collapse of wave
function. Let us represent the
coherent state as infinity since, if the
Hilbert space is separable, then any
its point is a coherent superposition
of a countable set of components.
The ‚collapse‛ represents as if a
descending avalanche from the
90. Nonstandard universum
Before a sketch of proof, its sense
and motivation: If that’s the case, the
axiom of foundation AF is available
just as the requirement for the
wave function to collapse from the
infinity as an avalanche since AF
forbids a smooth, continuous, infinite
lowering, sinking. It would be an
equivalent of the AF negation. A
smooth, continuous, infinite process
of lowering admits and even
91. Nonstandard universum
A note: Let us accept now the AF
negation, and consequently , a
smooth reversibility between
coherent and ‚collapsed‛ state.
Then: P = Ps - Pr, where Ps is the
probability from the coherent
superposition to a given value, and
Pr is the probability of reversible
process. So that the quantum
mechanical probability attached to
92. Nonstandard universum
A Minkowski operator is compact if
and only if it is a compact Hilbert
operator. A sketch of proof:
Wave function Y: RR RR
Hilbert space: {RR} {RR}
Hilbert operators:
{RR} {RR} {RR} {RR}
Using the isomorphism of Möbius and
Lorentz group as follows:
93. Nonstandard universum
{RR} {RR} {RR} {RR}
(the isomorphism)
{RR R}R {RR R}R:
i.e. Minkowski space operators.
The sense of introducing of
nonstandard infinitesimals by
compact Hilbert operators is for
them to be invariant towards
(straight and inverse)
94. Nonstandard universum
A little comment on the theorem:
A Minkowski operator is compact if
and only if it is a compact Hilbert
operator
Defining nonstandard infinitesimals
as compact Hilbert operators we
are introducing infinitesimals being
able to serve both such ones of the
transition between Minkowski and
Hilbert space (the apparatus and the
95. Nonstandard universum
A little more comment on the
theorem:
Let us imagine those infinitesimals,
being operators, as sells of phase
space: they are smoothly decreasing
from the minimal cell of the
apparatus phase space via and
beyond the axiom of foundation to
zero, what is the phase space sell of
the microobject. That decreasing is
96. Nonstandard universum
A little more comment on the
theorem:
Hamiltonian describes a system by
two independent linear systems of
equalities [as if towards the
reference frame both of the
apparatus (infinity) and of
microobject (finiteness)]
Lagrangian does the same by a
nonlinear system of equalities [the
97. Nonstandard universum
A little more comment on the
theorem:
Jacobian describes the bifurcation,
two-forked direction(s) from a
nonlinear system to two linear
systems when the one united,
common description is already
impossible and it is disintegrating to
two independent each of other
descriptions
98. Nonstandard universum
A few slides are devoted to
alternative ways for
nonstandard infinitesimals to be
introduced:
- smooth infinitesimal analysis
- surreal numbers.
Both the cases are inappropriate
to our purpose or can be
interpreted too close-ly or even
99. Nonstandard universum
‚Intuitively, smooth infinitesimal
analysis can be interpreted as
describing a world in which lines are
made out of infinitesimally small
segments, not out of points. These
seg-ments can be thought of as being
long enough to have a definite
direction, but not long enough to be
curved. The construction of
discontinuous functions fails because
a function is identified with a curve,
100. Nonstandard universum
‚We can imagine the intermediate
value theorem's failure as
resulting from the ability of an
infinitesimal segment to
straddle a line. Similarly, the
Banach-Tarski paradox fails
because a volume cannot be
taken apart into points‛
(Wikipedia, ‚Smooth infinitesimal
101. Nonstandard universum
The infinitesimals x in smooth
infinitesimal analysis are
nilpotent (nilsquare): x2=0
doesn’t mean and require that x
is necessarily zero. The law of
the excluded middle is denied:
the infinitesimals are such a
middle, which is between zero
and nonzero. If that’s the case
102. Nonstandard universum
The smooth infinitesimal analysis
does not satisfy our
requirements even only because
of denying the axiom of choice or
the Banach - Tarski paradox. But I
think that another version of
nilpotent infinitesimals is
possible, when they are an
orthogonal basis of Hilbert
space and the latter is being
103. Nonstandard universum
By introducing as zero divisors,
the infinitesimals are interested
because of possibility for the
phase space sell to be zero still
satisfying uncertainty. It means
that the bifurcation of the initial
nonlinear reference frame to
two linear frames
correspondingly of the
104. Nonstandard universum
The infinitesimals introduced as
surreal numbers unlike
hyperreal numbers (equal to
Robinson’s infinitesimals):
Definition: ‚If L and R are two
sets of surreal numbers and no
member of R is less than or
equal to any member of L then {
105. Nonstandard universum
About the surreal numbers:They
are a proper class (i.e. are not a
set), ant the biggest ordered
field (i.e. include any other
field). Comparison rule: ‚For a
surreal number x = { XL | XR } and
y = { YL | YR } it holds that x ≤ y if
and only if y is less than or
equal to no member of XL, and no
106. Nonstandard universum
Since the comparison rule is
recursive, it requires finite or
transfinite induction . Let us now
consider the following subset N
of surreal numbers: All the
surreal numbers S 0. 2N has to
contain all the well ordered
falling sequences from the
bottom of 0. The numbers of N
from the kind
107. Nonstandard universum
For example, we can easily to
define our initial problem in
their terms:
Let and be:
= {q: q {N | 0}}
= {w: w {0 | 0 N}}
Our problem is whether and
co-incide or not? If not, what is
power of ? Our hypothesis
is: the ans-wer of the former
108. Nonstandard universum
That special axiom set includes:
the axiom of choice and a
negation of the generalized
continuum hypothesis (GCH). Since
the axiom of choice is a
corollary from ZF+GCH, it implies
a negation of ZF, namely: a
negation of the axiom of
foundation AF in ZF. If ZF+GCH is the
case, our problem does not arise
109. Nonstandard universum
However a permission and
introducing of the infinite
degressive sequences , and
consequently, a AF negation is
required by quantum
information, or more
particularly, by a discussing
whether Hilbert and Minkowski
space are equivalent or not, or
more generally, by a considering
110. Nonstandard universum
Comparison between ‚standard‛
and nonstandard infinitesimals.
The‚standard‛ infinitesimals
exist only in boundary
transition. Their sense
represents velocity for a point-
focused sequence to converge to
that point. That velocity is the
ratio between the two neighbor
intervals between three discrete
111. Nonstandard universum
More about the sense of ‚standard‛
infinitesimals: By virtue of the axiom
of choice any set can be well
ordered as a sequence and thereby
the ratio between the two neighbor
intervals between three discrete
successive points of the sequence in
question is to exist just as before: in
the proper case of series. However
now, the ‚neighbor‛ points of an
arbitrary set are not discrete and
112. Nonstandard universum
Although the ‚neighbor‛ points of an
arbit-rary set are not discrete, and
consequently, the intervals between
them are zero, we can recover as
if ‚intervals‛ between the well-
ordered as if ‚discrete‛
neighbor points by means of
nonstandard infini-tesimals. The
nonstandard infinitesimals are
such intervals. The representation
113. Nonstandard universum
But the ratio of the neighbor
intervals can be also considered
as probability, thereby the
velocity itself can be inter-
preted as such probability as
above. Two opposite senses of a
similar inter-pretation are
possible: 1) about a point
belonging to the sequence: as
much the velocity of convergence
114. Nonstandard universum
2) about a point not belonging to
the sequence: as much the
velocity of convergence is higher
as the probability of a point out
of the series in question to be
there is less; i.e. the sequence
thought as a process is steeper,
and the process is more
nonequilibrium, off-balance,
dissipative while a balance,
115. Nonstandard universum
The same about a cell of phase
space:
The same can be said of a cell of
phase space: as much a process is
steeper, and the process is more
nonequilibrium, off-balance,
dissipative as the probability of
a cell belonging to it is higher
while a balance, equilibrium,
116. Nonstandard universum
Our question is how the
probability in quantum
mechanics should be interpre-
ted? A possible hypothesis is: the
pro-babilities of non-
commutative, comple-mentary
quantities are both the kinds
correspondingly and
interchangeably.
For example, the coordinate
117. Nonstandard universum
The physical interpretation of
the velo-city for a series to
converge is just as velocity of
some physical process. If the
case is spatial motion, then the
con-nection between velocity and
probability is fixed by the
fundamental constant c:
118. Nonstandard universum
The coefficients , from the
definition of qubit can be
interpreted as generalized,
complex possibilities of the
coefficients , from relativity:
Qubit: Relativity:
2+2=1
= (1-) 1/2
|0+|1 = q =v/c
119. Nonstandard universum
The interpretation of the ratio
between nonstandard
infinitesimals both as velocity
and as probability. The ratio
between ‚stanadard‛
infinitesimals which exist only in
boundary transit
120. Nonstandard universum
But we need some interpretation
of complex probabilities, or,
which is equi-valent, of complex
nonstandard neigh-borhoods. If
we reject AF, then we can
introduce the falling, descending
from the infinity, but also
infinite series as purely,
properly imaginary nonstandard
neighborhoods: The real
121. Nonstandard universum
After that, all the complex
probabilities are ushered in
varying the ties, ‚hyste-reses‛
‚up‛ or ‚down‛ between two
well ordered neighbor standard
points. Wave function being or
not in separable Hilbert space
(i.e. with countable or non-
countable power of its
components) is well interpreted
122. Nonstandard universum
Consequently, there exists one
more bridge of interpretation
connecting Hilbert and 3D or
Minkowski space.
What do the constants c and h
inter-pret from the relations
and ratios bet-ween two
neighbor nonstandard inter-
vals? It turns out that c
123. Nonstandard universum
And what about the constant h?
It guarantees on existing of: both
the sequences, both the
nonstandard neighborhoods ‚up‛
and ‚down‛. It is the unit of the
central symmetry transforming
between the nonstandard
neighborhoods ‚up‛ and ‚down‛
of any standard point h като площ
124. Nonstandard universum
And what about the constant h? It
gua-rantees on existing of: both
the sequen-ces, both the
nonstandard neighbor-hoods
‚up‛ and ‚down‛. It is the unit of
the central symmetry
transforming between the
nonstandard neighborhoods ‚up‛
and ‚down‛ of any stan-dard
125. Nonstandard universum
One more interpretation of h: as
the square of the hysteresis
between the ‚up‛ and the ‚down‛
neighborhood between two
standard points. Unlike standard
continuity a parametric set of
nonstandard continuities is
available. The parameter g =
Dp/Dx = Dm/Dt =
126. Nonstandard universum
One more interpretation of h:
The sense of g is intuitively very
clear: As more points ‚up‛ and
‚down‛ are common as both the
hysteresis branches are closer.
So the standard continuity turns
out an extreme peculiar case of
nonstan-dard continuity, namely
all the points ‚up‛ and ‚down‛
are common and both the
127. Nonstandard universum
By means of the latter
interpretation we can interpret
also phase space as non-
standard 3D space. Any cell of
phase space represents the
hysteresis between 3D points
well ordered in each of the
three dimensions. The connection
bet-ween phase space and Hilbert
128. Nonstandard universum
What do the constants c and h
interpret as limits of a phase
space cell deformation?
c.1.dx dy h.dx
Here 1 is the unit of curving
[distance x mass]
129. Forthcoming in 2nd part:
1. Motivation
2. Infinity and the axiom of choice
3. Nonstandard universum
4. Continuity and continuum
5. Nonstandard continuity
between two infinitely close
standard points
6. A new axiom: of chance
7. Two kinds interpretation of
130. CONTINUITY AND CONTINUUM
IN NONSTANDARD UNIVERSUM
Vasil Penchev
Institute for Philosophical Research
Bulgarian Academy of Science
E-mail: vasildinev@gmail.com
Professional blog:
http://www.esnips.com/web/vasilpenchevsnews
That was all of 1 st part
Thank you for your