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
In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical...In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of counting. Vertically it is continuous, horizontally it is discrete. While Frege, Tarski, Cantor, Gödel and the Vienna Circle tried to derive the higher dimension from the lower, a procedure that always leads to new contradictions and antinomies (Tarski, Russell), I take the opposite approach here, in which I derive the lower dimension from the higher. This perspective seems to fail because Tarski, Russell, Wittgenstein, and especially the Vienna Circle have shown that the completeness of the absolute itself is logically contradictory. For this reason, we agree with Hegel in assuming that we can never fully comprehend the Absolute, but only its particular manifestations—otherwise we would be putting ourselves in the place of the Absolute, or even God. Nevertheless, we can understand the Absolute in its particular expressions, as I will show with the modest example of the triangle proof of the combined horizontal and vertical countability of the real numbers, which I developed in rejection of Cantor’s diagonal proof. .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.
文摘In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of counting. Vertically it is continuous, horizontally it is discrete. While Frege, Tarski, Cantor, Gödel and the Vienna Circle tried to derive the higher dimension from the lower, a procedure that always leads to new contradictions and antinomies (Tarski, Russell), I take the opposite approach here, in which I derive the lower dimension from the higher. This perspective seems to fail because Tarski, Russell, Wittgenstein, and especially the Vienna Circle have shown that the completeness of the absolute itself is logically contradictory. For this reason, we agree with Hegel in assuming that we can never fully comprehend the Absolute, but only its particular manifestations—otherwise we would be putting ourselves in the place of the Absolute, or even God. Nevertheless, we can understand the Absolute in its particular expressions, as I will show with the modest example of the triangle proof of the combined horizontal and vertical countability of the real numbers, which I developed in rejection of Cantor’s diagonal proof. .
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.
