An Analysis of Zeno’s Paradox and Non-measurable Sets Based on Dialectical Infinity

The Problem of Continuity and Discreteness is the basic problem of philosophy and mathematics. For a long time, there is no clear understanding of this problem, which leads to the stagnation of the problem, because the essence of the problem is a problem of finity and infinity. The essence of the philosophical thought on which the mathematical definition of “line segment is composed of dots” is the idea of actual infinity, and geometric dot is equivalent to algebraic zero in terms of measure properties. In view of the above contradictions, this paper presents two solutions satisfying both the philosophical and mathematical circles based on the view of dialectical infinity, and the authors make a deep analysis of Zeno’s paradox and the non-measurable set based on both solutions.


Analysis of the Problem
On the mathematical definition of point (or dot), it is considered that a point has no measure and a line segment has measure, but on the other hand, mathematics stipulates that 'a line segment is composed of points', which leads to an inherent irreconcilable contradiction in mathematics. The point of mathematics is an absolute divided thing (which cannot be further divided). The line segment is composed of infinite points, which is composed of many points of actual infinity, thus a definition of actual infinite property. This idea of actual infinity inevitably leads to the lack of qualitative unity between line segment and point (the essence of actual infinity is to separate the infinite process from the result qualitatively), leading to the internal inconsistency between the point and the line segment, which leads to the irreconcilable contradiction, so that the line segment is composed of points is contradictory and absurd in philosophy.
Point, as a purely mathematical abstraction, is zero in length (measure), is nothingness and an imaginary thing. This kind of geometric pure point, like the pure zero in algebra, is an absolute nothingness (emptiness), which has no inherent connection with space and continuity, and has no qualitative unity; therefore, more points can only constitute a point of nothingness, can not constitute continuity. Line segment and point (or set of points) are two completely different concepts, no one can represent the other. There can be countless points of nothingness on the line segment, but what does this have to do with the line segment? Therefore, the line segment is not composed of points, and the face cannot be composed of straight lines and the space cannot be composed of faces.
However, if the point is regarded as an impure abstract, if it has a length, it represents a process rather than a stationary object, it is a variable; it can be seen as a limit process of shrinking a line segment, thus a line segment with infinite small measure, then point and line segment will have qualitative unity, and the line segment will become understandable by the composition of points.
Quality is the internal stipulation of things, and quantity is the external stipulation of things. (Zhang & Zhuang, 2019) holds that real infinity (or true infinity) is the connection and unity of the inner quality of bad infinity, so real infinity represents and reflects the quality of bad infinity; infinity is the unity of quality and quantity, bad (or evil) infinity represents quantity of infinity (motion), real infinity represents quality (law, commonness or connection) of infinity. The dialectical infinity view holds that the actual infinity separates the finite from the infinite, and looks at things from the point of view of stillness rather than movement, which has an internal irreconcilable contradiction, and is a metaphysical idealistic infinite view. As a mathematical abstract concept, the inherent quality of a line segment is continuity and if it is an infinite (composed of infinite divided forever, that is, divided into measured points, which is a process of bad infinity and inexhaustible. Therefore, we must give up the expression and definition of 'line segment is composed of points', which is an inherent contradiction, and give up the definition of actual infinite property.

Solutions
Looking at the analysis of the major philosophers and mathematicians in history (Zhang & Zhou, 2021), two solutions have surfaced. For the relationship between points and line segments, we have two acceptable understandings, both of which can solve the problem of continuity and discreteness well.
One is that "a line segment is composed of points", but it must be recognized that point has a measure of infinitesimal (variable), that is, a point has a measure that is both zero and not zero, and the point is still essentially a line segment. The basis of this solution is to confirm the qualitative unity of point and line segment. Therefore, the point is still an infinite process rather than an end point, that is, this point is not the point in which the measure in mathematics is purely zero. In this way, the line segment consists of points becomes an infinite process, an endless process. This accords with the understanding of dialectical infinity, and there is no inherent contradiction. Hegel and Peirce proposed such a solution.
This solution shows that the measure (quantity) of a measured point is a unity of being and not being, a changing infinitely small quantity, and a variable. Macroscopically, the measure of single point is zero, while relative microscopic, the measure of a single point is not zero, but an infinite small quantity (variable), which embodies an infinite process. Therefore, the measure of an infinite number of such points, macroscopically, may be zero or not, which requires a clear qualitative relationship between the infinity and the segment.
Another kind of understanding is to oppose "a line segment consists of points", that is, the line segment is not composed of points, the line segment itself is a basic mathematical abstract object with measure (that is, continuity as a basic abstract object), the line segment has nothing to do with points, and has nothing to do with infinite and infinite processes. Wittgenstein and other masters. This concept extends to the understanding of number, that is, the set of real numbers is a mark set, the number is the mark of the quantity, and the number corresponds to the line segment on the number axis. As Mr.WangHao said, "The measurement of length and volume is a combination of arithmetic and geometry, applying various units to calculate a number. As with the solution of the equation, this is a natural way to produce fractions and irrational numbers. The concept of real numbers arises from measurements that require absolute precision and, more precisely, can be infinitely improved."

The Number Itself Does Not Reflect Continuity
Modern mathematics equates the set of real numbers with the continuum (line segment), so that each real number corresponds to a mathematical point, and the continuity of real numbers is used to reflect the continuity of the line segment. However, the number itself, like the point, cannot represent and reflect continuity, the essence of the number is representing separation.
In Hegel's view, the number itself cannot explain the continuity, the essence of the number reflects the separation; Hegel believes that quantity includes two links: continuity and dispersion, which is unified by quantity, that is, continuity and dispersion are dialectical unity through quantity.
The essence of quantity originates from measurement, which is the calculation of length naturally, but not continuity itself. Number is the embodiment of quantity, so it is called quantity; Hegel believes that "number is the absolute determinateness of quantity" (Hegel, 1969, p. 212), Engels believed that "number is the purest quantitative determination that we know" (Engels, p. 262). Therefore, numbers can represent both discrete quantities, such as numbers, counts, and continuous quantities, such as units, scales (lengths). When the number represents a continuous quantity, it is based on the unit scale (a continuous) and the number is the measurement of the continuous system. Therefore, the number hides the continuity through the unit and discretizes the continuity object, so the number itself cannot explain the continuity, and the essence of quantity is to embody the separation.
Hegel emphasizes the unity of continuity and separation (discreteness). He points out that, "Conversely, in discrete magnitude continuity is not to be overlooked; this moment is, as has been shown, the one as unity." (Hegel, 1969, p. 200); That is, the basis of number is unit, scale, so the fundamental characteristic of number is measurement. An essence of unit, scale is a quantum, is "a quantum, to which a determinate being or a quality is attached." (Hegel, 2015, p. 172), is the unity of quality and quantity. Hegel believes that the essence of quantity is separation, which does not directly reflect continuity. He states: "Immediate quantity is continuous magnitude. But quantity is not an immediate at all; immediacy is a determinateness the sublatedness of which is quantity itself. It is, therefore, to be posited in the determinateness immanent in it, and this is the one. Quantity is discrete magnitude." (Hegel, 1969, p. 200) Hegel introduced the concept of number in detail, and pointed out that the number from the separation link is the sum, from the continuous link is the unit. Number embodies many, which can represent both continuous and discrete quantities. One as the embodiment of unit, sublation of continuity, that is, one does not reflect continuity and unit itself reflects continuity; The number is based on unit (one), and the number has continuity in unit, which deeply reveals that the number hides continuity through unit. Hegel states, "Quantum, which to begin with is quantity with a determinateness or limit in general is, in its complete determinateness, number." (Hegel, 1969, p. 202), "Number exhibits the development and perfect character of the Quantum. Like the One, the medium in which it exists, Number involves two qualitative factors or functions; Annumeration or Sum, which depends on the discrete influence, and Unity, which depends on continuity." (Hegel, 2015, p. 163) because the motion is gone there (the essence of the actual infinity is a denial of the motion), so the arrow flying is still. As Aristotle put it, "this conclusion is caused by treating time as being synthesized by presents." (Aristotle,p. 192); He pointed out that "nothing can move in the present" (Aristotle,p. 172), "the movement of moving things and the stillness of still things must be for a while." (Aristotle,p. 173) Time is reflected by movement. Only when we feel the change of movement can we feel time. Movement must be a movement in time.
The second explanation of Dialectical Infinity (that is, the line segment is independent of the point, time is independent of moment): That is, time has nothing to do with the point of present (moment); without the moment of present, the problem itself does not exist.

Analysis of non Lebesgue-measurable Sets
We begin by constructing a non-measurable set. For the rational number set Q, the following family of sets is defined: , thereinto x Q x R   , and R represents the set of real numbers. The family of sets comes from Q translation arbitrarily to left or to right (called as the coset of Q), each of which is equal to Q and is dense in R. These cosets have the following properties: One is that different The second is that the union of all the cosets is exactly R, i.e., Below, we will specifically construct the non-measurable set. Since each coset x Q  is dense in R, . One element is now selected from each of all these disjoint intersection X (from the Axiom of Choice), and these elements form a new set V which is a non-measurable set. Generally speaking, we decompose interval [0,1] into disjoint countable sets of infinitely many (uncountable), and the measure of X is equal to 0, i.e., ( ) 0 X   , because the measure of a countable set is 0. Select one element from each of these disjoint sets X and form a new set V, which is a non-measurable set that has the same cardinal number as the interval [0,1] and is an uncountable set.
Analysis of the measure of V. On the one hand, if ( ) 0 V   and the interval [0,1] is composed of such sets V of countable many (proof omitted), this will cause the measure of [0,1] to be also 0.
 , on the other hand, this will cause the measure of the union of these sets V (i.e., interval [0,1]) to become infinite, which is obviously not possible. Therefore, V is a non-measurable set.
The fundamental reason why V becomes a non-measurable set lies in the basic mathematical contradiction that "a line segment is composed of points". The existence of a non-measurable set deeply reflects and reveals the internal contradictions of basic mathematics.
The first explanation of Dialectical Infinity (that is, a point is essentially a line segment): In this case, a single dot has an infinitesimal measure (variable), namely, the dot has a measure that is both zero and not zero. Similarly, the measure of countable set X is an infinitesimal measure (variable). The interval [0,1] consists of countable infinitely many V sets, and each V set is mutually equivalent, so, the measure of V is also an infinitesimals, both zero and not zero; countable infinitely many infinitesimals constitute measure 1. It is only that the infinitesimals here is different from the infinitesimals of the measure of set X, and the relationship between them is determined by the intrinsic connection between their properties of infinity.
Hegel holds that infinity are comparable and can be compared only in the sense of real infinity, that is, in the sense of the qualitative determinateness and qualitative connectivity of infinity. The two sets of V and X (namely, two infinity) have an internal qualitative connection, the cardinal number of V is equivalent to the one of the interval [0,1] (i.e., the cardinal number of R), the cardinal number of X is the same as Q, R and Q are the relation of the power-set property, so there is an internal qualitative connection between V and X, so their measures can be compared in size. Therefore, although both the measures of V and X are infinitely small, they are at different levels of infinite; the measure of V is an infinitely small relative to the measure 1 of the interval [0,1], while the measure of X is a lower rank infinitely small.
Thus, under the condition that the point is still a line segment, V is still a measurable set. While the measures of all sets are relative, namely the measure of the interval [0,1] is also relative rather than absolute.
The second explanation of Dialectical Infinity (that is, the line segment is independent of the point, time is independent of moment): Since the line segment has nothing to do with the point, there is no so-called measure in any of the point sets, and the formulation of the non-measurable set is meaningless.