A previous paper showed that the real numbers between 0 and 1 could be represented by an infinite tree structure, called the ‘infinity tree’, which contains only a countably infinite number of nodes and arcs. This p...A previous paper showed that the real numbers between 0 and 1 could be represented by an infinite tree structure, called the ‘infinity tree’, which contains only a countably infinite number of nodes and arcs. This paper discusses how a finite-state Turing machine could, in a countably infinite number of state transitions, write all the infinite paths in the infinity tree to a countably infinite tape. Hence it is argued that the real numbers in the interval [0, 1] are countably infinite in a non-Cantorian theory of infinity based on Turing machines using countably infinite space and time. In this theory, Cantor’s Continuum Hypothesis can also be proved. And in this theory, it follows that the power set of the natural numbers P(ℕ) is countably infinite, which contradicts the claim of Cantor’s Theorem for the natural numbers. However, this paper does not claim there is an error in Cantor’s arguments that [0, 1] is uncountably infinite. Rather, this paper considers the situation as a paradox, resulting from different choices about how to represent and count the continuum of real numbers.展开更多
Refs 1 and 2 provide the definition of the concepts of‘potential infinity’(poi)and actual infinity(aci);Ref 3 discusses and verifies that poi and aci are a pair of contradictory opposites without intermediate(p,-p)....Refs 1 and 2 provide the definition of the concepts of‘potential infinity’(poi)and actual infinity(aci);Ref 3 discusses and verifies that poi and aci are a pair of contradictory opposites without intermediate(p,-p).The second part of this paper,i.e.,§2,further discusses the manners in which a variable x approaches infinitely to its limit x0 using the poi and aci methods and concludes that,in any system compatible with both poi and aci, the two approaching manners are also a pair of contradictory opposites without intermediate (A,-A).Finally,on the basis of this conclusion,we reexamine the fundamental question of Leibniz’s Secant and Tangent Lines in calculus and the limit theory and offer our analysis and raise new questions.展开更多
Abstract： Ref [5] provides a logical-mathematical explanation of the incompatibility ofLeibniz＇s secant and tangent lines in medium logic. However, the expression （＊）（△y/△x） ismeaningful and dy/dx is the tang...Abstract： Ref [5] provides a logical-mathematical explanation of the incompatibility ofLeibniz＇s secant and tangent lines in medium logic. However, the expression （＊）（△y/△x） ismeaningful and dy/dx is the tangent slope） derived from ⑦ and ⑧ in §4 of Ref [5] is unimaginablewithin the framework of two-valued logic, why shouldn＇t the same conflicting concluslon be reached in the medium logic calculus？ This paper has subjected these questions to careful logical analysis, and approached them from the perspective of logical mathematics. As the two approaches have led to the identical conclusion, the paper thereby rigorously and thoroughlv answers these questions.展开更多
This paper discusses how the infinite set of real numbers between 0 and 1 could be represented by a countably infinite tree structure which would avoid Cantor’s diagonalization argument that the set of real numbers i...This paper discusses how the infinite set of real numbers between 0 and 1 could be represented by a countably infinite tree structure which would avoid Cantor’s diagonalization argument that the set of real numbers is not countably infinite. Likewise, countably infinite tree structures could represent all real numbers, and all points in any number of dimensions in multi-dimensional spaces. The objective of this paper is not to overturn previous research based on Cantor’s argument, but to suggest that this situation may be treated as a definitional or axiomatic choice. This paper proposes a “non-Cantorian” branch of cardinality theory, representing all these infinities with countably infinite tree structures. This approach would be consistent with the Continuum Hypothesis.展开更多
An H infinity(H∞)controller for a sandwiched maglev positioning stage is proposed.The maglev positioning stage has a special structure:a sandwiched maglev stage,consisting of repulsive linear motors and attractive li...An H infinity(H∞)controller for a sandwiched maglev positioning stage is proposed.The maglev positioning stage has a special structure:a sandwiched maglev stage,consisting of repulsive linear motors and attractive linear motors,which have better levitation performance.Forces on the sandwiched maglev stage are analyzed and modeled.The positioning controller is designed based on the feedback linearized model with a dynamic damping system.The design of the H infinity controller for stage positioning is derived as a series of linear matrix inequalities(LMIs)which are efficiently solved in Matlab.The proposed controller and its effectiveness is demonstrated compared to PID method.展开更多
From the perspective of potential infinity (poi) and actual infinity, Ref [4] has confirmed that poi and aci are in 'unmediated opposition' (P,﹁P ) whether in ZFC or not; it has further been proved that the m...From the perspective of potential infinity (poi) and actual infinity, Ref [4] has confirmed that poi and aci are in 'unmediated opposition' (P,﹁P ) whether in ZFC or not; it has further been proved that the manners in which a variable infinitely approaches its limit also satisfy the law of intermediate exclusion. With these results as theoretical bases, this paper attempts to provide an accurate and strict logical-mathematical interpretation of the incompatibility of Leibniz's secant and tangent lines in the medium logic system from the perspective of logical mathematics.展开更多
This paper presents the hypermedia data model based on the infinity RS image information system we have developed.The hypermedia data model consists of different semantic units called nodes,and the associations betwee...This paper presents the hypermedia data model based on the infinity RS image information system we have developed.The hypermedia data model consists of different semantic units called nodes,and the associations between nodes are called links.This paper proposes three kinds of nodes (interior node,physical node and complex node) and two kinds of links (plane network structure link,hyper_cube network structure links).The hypermedia information system,based on the model and the basic data layer (the infiniy RS image),represents a digital globe.An approach to the “Getting Lost in the Hyper_space” problem is presented.The approach using the hypermedia data model is an efficient way of handling a large number of RS images in various geographical information systems.展开更多
文摘A previous paper showed that the real numbers between 0 and 1 could be represented by an infinite tree structure, called the ‘infinity tree’, which contains only a countably infinite number of nodes and arcs. This paper discusses how a finite-state Turing machine could, in a countably infinite number of state transitions, write all the infinite paths in the infinity tree to a countably infinite tape. Hence it is argued that the real numbers in the interval [0, 1] are countably infinite in a non-Cantorian theory of infinity based on Turing machines using countably infinite space and time. In this theory, Cantor’s Continuum Hypothesis can also be proved. And in this theory, it follows that the power set of the natural numbers P(ℕ) is countably infinite, which contradicts the claim of Cantor’s Theorem for the natural numbers. However, this paper does not claim there is an error in Cantor’s arguments that [0, 1] is uncountably infinite. Rather, this paper considers the situation as a paradox, resulting from different choices about how to represent and count the continuum of real numbers.
基金Supported by the Open Fund of the State Key Laboratory of Software Development Environment(SKLSDE-2011KF-04)Supported by the Beihang University and by the National High Technology Research and Development Program of China(863 Program)(2009AA043303)
文摘Refs 1 and 2 provide the definition of the concepts of‘potential infinity’(poi)and actual infinity(aci);Ref 3 discusses and verifies that poi and aci are a pair of contradictory opposites without intermediate(p,-p).The second part of this paper,i.e.,§2,further discusses the manners in which a variable x approaches infinitely to its limit x0 using the poi and aci methods and concludes that,in any system compatible with both poi and aci, the two approaching manners are also a pair of contradictory opposites without intermediate (A,-A).Finally,on the basis of this conclusion,we reexamine the fundamental question of Leibniz’s Secant and Tangent Lines in calculus and the limit theory and offer our analysis and raise new questions.
文摘Abstract： Ref [5] provides a logical-mathematical explanation of the incompatibility ofLeibniz＇s secant and tangent lines in medium logic. However, the expression （＊）（△y/△x） ismeaningful and dy/dx is the tangent slope） derived from ⑦ and ⑧ in §4 of Ref [5] is unimaginablewithin the framework of two-valued logic, why shouldn＇t the same conflicting concluslon be reached in the medium logic calculus？ This paper has subjected these questions to careful logical analysis, and approached them from the perspective of logical mathematics. As the two approaches have led to the identical conclusion, the paper thereby rigorously and thoroughlv answers these questions.
文摘This paper discusses how the infinite set of real numbers between 0 and 1 could be represented by a countably infinite tree structure which would avoid Cantor’s diagonalization argument that the set of real numbers is not countably infinite. Likewise, countably infinite tree structures could represent all real numbers, and all points in any number of dimensions in multi-dimensional spaces. The objective of this paper is not to overturn previous research based on Cantor’s argument, but to suggest that this situation may be treated as a definitional or axiomatic choice. This paper proposes a “non-Cantorian” branch of cardinality theory, representing all these infinities with countably infinite tree structures. This approach would be consistent with the Continuum Hypothesis.
基金Supported by the National Natural Science Foundation of China(51375052)
文摘An H infinity(H∞)controller for a sandwiched maglev positioning stage is proposed.The maglev positioning stage has a special structure:a sandwiched maglev stage,consisting of repulsive linear motors and attractive linear motors,which have better levitation performance.Forces on the sandwiched maglev stage are analyzed and modeled.The positioning controller is designed based on the feedback linearized model with a dynamic damping system.The design of the H infinity controller for stage positioning is derived as a series of linear matrix inequalities(LMIs)which are efficiently solved in Matlab.The proposed controller and its effectiveness is demonstrated compared to PID method.
基金Supported by the Open Fund of the State Key Laboratory of Software Development Environment(SKLSDE-2011KF-04)Supported by the National High Technology Research and Development Program of China (863 Program)(2009AA043303)
文摘From the perspective of potential infinity (poi) and actual infinity, Ref [4] has confirmed that poi and aci are in 'unmediated opposition' (P,﹁P ) whether in ZFC or not; it has further been proved that the manners in which a variable infinitely approaches its limit also satisfy the law of intermediate exclusion. With these results as theoretical bases, this paper attempts to provide an accurate and strict logical-mathematical interpretation of the incompatibility of Leibniz's secant and tangent lines in the medium logic system from the perspective of logical mathematics.
文摘This paper presents the hypermedia data model based on the infinity RS image information system we have developed.The hypermedia data model consists of different semantic units called nodes,and the associations between nodes are called links.This paper proposes three kinds of nodes (interior node,physical node and complex node) and two kinds of links (plane network structure link,hyper_cube network structure links).The hypermedia information system,based on the model and the basic data layer (the infiniy RS image),represents a digital globe.An approach to the “Getting Lost in the Hyper_space” problem is presented.The approach using the hypermedia data model is an efficient way of handling a large number of RS images in various geographical information systems.
