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610There is No Standard Model of ZFC and ZFC_2. Part I.Journal of Advances in Mathematics and Computer Science 2 (26): 1-20. 2017.In this paper we view the first order set theory ZFC under the canonical frst order semantics and the second order set theory ZFC_2 under the Henkin semantics. Main results are: (i) Let M_st^ZFC be a standard model of ZFC, then ¬Con(ZFC + ∃M_st^ZFC ). (ii) Let M_stZFC_2 be a standard model of ZFC2 with Henkin semantics, then ¬Con(ZFC_2 +∃M_stZFC_2). (iii) Let k be inaccessible cardinal then ¬Con(ZFC + ∃κ). In order to obtain the statements (i) and (ii) examples of the inconsistent c…Read more
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449Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance (edited book)Book Publisher International. 2019.This book deals with Colombeau solutions to Einstein field equations in general relativity: Gravitational singularities, distributional SAdS BH spacetime-induced vacuum dominance. This book covers key areas of Colombeau nonlinear generalized functions, distributional Riemannian, geometry, distributional schwarzschild geometry, Schwarzschild singularity, Schwarzschild horizon, smooth regularization, nonsmooth regularization, quantum fields, curved spacetime, vacuum fluctuations, vacuum dominance …Read more
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426Inconsistent Countable Set in Second Order ZFC and Nonexistence of the Strongly Inaccessible CardinalsBritish Journal of Mathematics and Computer Science 9 (5): 380-393. 2015.In this article we derived an important example of the inconsistent countable set in second order ZFC (ZFC_2) with the full second-order semantics. Main results: (i) :~Con(ZFC2_); (ii) let k be an inaccessible cardinal, V is an standard model of ZFC (ZFC_2) and H_k is a set of all sets having hereditary size less then k; then : ~Con(ZFC + E(V)(V = Hk)):
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385EPRB Paradox Resolution.Bell inequalities revisited. (edited book)Amazon. 2019.This book is devoted to the presentation of the new quantum mechanical formalism based on the probability representation of quantum states. In the 20s and 30s it became evident that some properties in quantum mechanics can be assigned only to the quantum mechanical system, but not necessarily to its constituents. This led Einstein, Podolsky and Rosen (EPR) to their remarkable 1935 paper where they concluded that quantum mechanics is not a complete theory of nature (EPR paradox). In order to avoi…Read more
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364Relevant first-order logic LP# and Curry’s paradox resolutionPure and Applied Mathematics Journal Volume 4, Issue 1-1, January 2015 DOI: 10.11648/J.Pamj.S.2015040101.12. 2015.In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventin…Read more
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349The Solution Cosmological Constant ProblemJournal of Modern Physics 10 (7): 729-794. 2019.The cosmological constant problem arises because the magnitude of vacuum energy density predicted by the Quantum Field Theory is about 120 orders of magnitude larger then the value implied by cosmological observations of accelerating cosmic expansion. We pointed out that the fractal nature of the quantum space-time with negative Hausdorff-Colombeau dimensions can resolve this tension. The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff ∗ Λ a…Read more
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346Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I)Journal of Advances in Mathematics and Computer Science 36 (2): 73-88. 2021.In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.
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337There is no standard model of ZFCJournal of Global Research in Mathematical Archives 5 (1): 33-50. 2018.Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st), (ii) let k be an inaccessible cardinal then ~Con(ZFC+∃k),[10],11].
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322Was Polchinski Wrong? Colombeau Distributional Rindler Space-Time with Distributional Levi-Cività Connection Induced Vacuum Dominance. Unruh Effect RevisitedJournal of High Energy Physics, Gravitation and Cosmology 2 (4): 361-440. 2018.The vacuum energy density of free scalar quantum field Φ in a Rindler distributional space-time with distributional Levi-Cività connection is considered. It has been widely believed that, except in very extreme situations, the influence of acceleration on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is wrong by showing that in a Rindler distributional background space-time with distributional Levi-Cività connection the vacuum energy of fr…Read more
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306Distributional SAdS BH Spacetime-Induced Vacuum DominanceJournal of Advances in Mathematics and Computer Science 13 (6): 1-54. 2016.This paper dealing with extension of the Einstein eld equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [1], [2], [3], [4], [5], [6], [7] and [32]. The regularizations of singularities presented in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Col…Read more
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290There is No Standard Model of ZFC and ZFC2. Part II.Advanced in Pure Mathematic 9 (9): 685-744. 2019.In this article we proved so-called strong reflection principles corresponding to formal theories Th which has omega-models or nonstandard model with standard part. An posible generalization of Lob’s theorem is considered.Main results are: (i) ConZFC Mst ZFC, (ii) ConZF V L, (iii) ConNF Mst NF, (iv) ConZFC2, (v) let k be inaccessible cardinal then ConZFC .
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289Generalized Löb’s Theorem.Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology.Pure and Applied Mathematics Journal (Vol. 4, No. 1-1): 1-5. 2015.
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244The EPR-B Paradox Resolution. Bell inequalities revisited.Journal of Physics: Conference Series, 1391 (1). 2019.One of the Bell's assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., the absence of the in uence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of a local realism. It is well known that the Bell's inequality was derived in its traditional form, without resorting to the hypothes…Read more
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212In contemporary mathematics, a Colombeau algebra of Colombeau generalized functions is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this. Remark 1.1.1.Such a multiplication of distributions has been a long time mistakenly believed to be impossible because of Schwartz’ impossibility result, which basically states that…Read more
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208New analytical approach for transition to slow 3-D turbulenceJournal of Physics: Conference Series 633 (1): 6. 2015.
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204New exact quasi-classical asymptotic beyond WKB approximation and beyond Maslov formal expansionJournal of Physics: Conference Series 633 (1). 2015.New exact quasi-classical asymptotic of solutions to the
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202Schrödinger’s Cat Paradox Resolution Using GRW Collapse Model: Von Neumann Measurement Postulate RevisitedJournal of Applied Mathematics and Physics 5 (2): 494-521. 2017.
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191Dark Matter NatureJournal of Physics: Conference Series 1391 (1). 2019.The cosmological constant problem arises because the magnitude of vacuum energy density predicted by quantum eld theory is about 120 orders of magnitude larger than the value implied by cosmological observations of accelerating cosmic expansion. We pointed out that the fractal nature of the quantum space-time with negative Hausdor- Colombeau dimensions can resolve this tension. The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cuto and ther…Read more
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191Generalized Löb’s Theorem. Strong Reflection Principles and Large Cardinal AxiomsAdvances in Pure Mathematics (3): 368-373. 2013.In this article, a possible generalization of the Löb’s theorem is considered. Main result is: let κ be an inaccessible cardinal, then ¬Con( ZFC +∃κ) .
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187Set theory INC# based on intuitionistic logic with restricted modus ponens rule (edited book)AP LAMBERT Academic Publishing (June 23, 2021). 2021.In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality. Similar results for paraconsistent set theories were obtained in author papers [13]-[16].
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187Novel explanation of the Active Galactic Nuclei. The Power Source of Quasars as a result of vacuum polarization by the gravitational singularities on the distributional BHs horizonJournal of Physics: Conference Series 1730 (1): 2-25. 2021.In this paper we argue that the current paradigm for AGN and quasars is essentially incomplete and a rivision is needed. Remind that the current paradigm for AGN and quasars is that their radio emission is explained by synchrotron radiation from relativistic electrons that are Doppler boosted through bulk motion. In this model, the intrinsic brightness temperatures cannot exceed 1011 to 1012 K. Typical Doppler boosting is expected to be able to raise this temperature by a factor of 10. The obser…Read more
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177Internal Set Theory IST# Based on Hyper Infinitary Logic with Restricted Modus Ponens Rule: Nonconservative Extension of the Model Theoretical NSAJournal of Advances in Mathematics and Computer Science 37 (7). 2022.The incompleteness of set theory ZF C leads one to look for natural nonconservative extensions of ZF C in which one can prove statements independent of ZF C which appear to be “true”. One approach has been to add large cardinal axioms.Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory T G or It is a nonconservative extension of ZF C and is obtained from other axiomatic set theories by the inclusion of Tarski’s axiom which implies …Read more
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169Non-archimedean analysis on the extended hyperreal line *R_d and the solution of some very old transcendence conjectures over the field Q.Advances in Pure Mathematics 5 (10): 587-628. 2015.In 1980 F. Wattenberg constructed the Dedekind completiond of the Robinson non-archimedean field and established basic algebraic properties of d [6]. In 1985 H. Gonshor established further fundamental properties of d [7].In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completiond in transcendental number theory w…Read more
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167Set Theory INC# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper inductive definitionsJournal of Advances in Mathematics and Computer Science 36 (4): 22. 2021.In this paper intuitionistic set theory INC# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.
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156Set Theory INC_{∞^{#}}^{#} Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part III).Hyper inductive definitions. Application in transcendental number theory.Journal of Advances in Mathematics and Computer Science 36 (8): 43. 2021.Main results are: (i) number e^{e} is transcendental; (ii) the both numbers e+π and e-π are irrational.
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155The Solution of the Invariant Subspace Problem. Part I. Complex Hilbert space.Journal of Advances in Mathematics and Computer Science 37 (10): 51-89. 2022.The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements independent of ZFC which appear to be "true". One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski- Grothendieck set theory TG [1]-[3] It is a non-conservative extension of ZFC and is obtaineed from other axiomatic set theories by the inclusion of Tarski's axiom which …Read more
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135Model λ(φ^2n )_4,n≥2 Quantum Field Theory: A Nonstandard Approach Based on Nonstandard Pointwise-Defined Quantum FieldsJournal of Physics: Conference Series 35. forthcoming.A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator φ(x,t) no longer a standard tempered operator-valued distribution, but a non-classical operator-valued function. We prove using this novel approach that the quantum field theory with Hamiltonian P(φ)_4 exists and that the corresponding C^* algebra of bounded observables satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the λ(φ^2n )_4,n≥2 quantum field the…Read more
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132In this paper paraconsistent first-order logic LP^{#} with infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth^{#} is discussed.Axiomatical system HST^{#}as paraconsistent generalization of Hrbacek set theory HST is considered
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129Einstein’s 1927 gedanken experiment revisitedJournal of Global Research in Mathematical Archives(JGRMA) 5 (7). 2018.In 1935, Einstein, Podolsky and Rosen (EPR) originated the famous “EPR paradox” [1]. This argument concerns two spatially separated particles which have both perfectly correlated positions and momenta, as is predicted possible by quantum mechanics. The EPR paper spurred investigations into the nonlocality of quantum mechanics, leading to a direct challenge of the philosophies taken for granted by most physicists.The EPR conclusion was based on the assumption of local realism, and thus the …Read more
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128Unruh effect revisitedJournal of Physics: Conference Series 1141 (1). 2018.The vacuum energy density of free scalar quantum field phgr in a Rindler distributional space-time with distributional Levi-Cività connection is considered. It has been widely believed that, except in very extreme situations, the influence of acceleration on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is wrong by showing that in a Rindler distributional background space-time with distributional Levi-Cività connection the vacuum energy of…Read more