Spacetime Divisibility

17 Spacetime divisibility Chapter of the book Infinity Put to the Test by Antonio León available HERE Abstract.-This chapter discusses the possibilities to divide finite intervals of space and time into infinitely many parts of a decreasing length. According to its negative conclusion, it also proposes the consideration of discrete spacetimes as an alternative to the currently assumed spacetime continuum. Keywords: infinite partitions of real intervals, theorem of the infinite partitions, spacetime divisibility, discrete spacetime. Introduction P342 In Chapter 16 it was proved that in the Euclidean space R3 any line* with two endpoints can only be divided into a finite number of parts of equal finite length. As a consequence, it was also proved that in the same Euclidean space the distance between any two of its points is always finite, and that all lines, whether open or closed, have a finite length. This chapter discusses the possibility of dividing any finite interval (of space or time) into an infinite number of parts of decreasing lengths (durations), which is the only way in which a finite length (duration) can presumably be divided into infinitely many parts. This would be the Aristotelian infinity by division [1, Books 3 and 6], and the result of the discussion is also the inconsistency of such infinite divisions. P343 Dividing an interval (a, b) into a given (finite or infinite) number of parts is to define a sequence of adjacent and disjoint parts such that: (a, b) = (a, x1 )[x1 , x2 )[x2 , x3 ) . . . [xn , b) (1) (a, b) = (a, y1 )[y1 , y2 )[y2 , y3 ) . . . (2) In the first case, equation (1), the division is finite (Theorem of the Finite Sets). It would be a partition of the interval (see P222). In the second, equation (2), the division would be infinite, without a last interval. It would be an ω-ordered segmentation (see P224). In such a case, the sequence of points hxi i defining the ω-segmentation will also be ω-ordered, and its limit will be the right endpoint b of the interval (a, b). P344 In the case of the ω-ordered segmentations of an open interval (a, b), the limit of the increasing (decreasing) sequence of points defining the partition is the right endpoint b (left endpoint a) of the interval. We know that between each of the points of the sequence and its limit there is always the same infinite number ℵo of points of the sequence (ω-asymmetry). Moreover, although the successive points approach their limit, none of them reach it. The limit point is not a point of the sequence. P345 Recall that, according to the definitions given in P222-P224, the collection of intervals: [x1 , x2 )[x2 , x3 )[x3 , x4 ) . . . [xω , xω+1 ) (3) is not a partition because the left endpoint xω of the last interval is not common to any other interval. It is also not a ω-segmentation because its ordinal is ω + 1. Therefore it is a (ω + 1)-segmentation. An example of ω-segmentation would be: [x1 , x2 )[x2 , x3 )[x3 , x4 ) . . . 1 (4) Divisibility of real intervals 2 Divisibility of real intervals P346 In P236-P241 of Chapter11 a real interval (the real interval (0, 1]) was divided into an infinite sequence of parts with a first and a last part, each part (except the first) having an immediate predecessor and an immediate successor (except the last). Of course, that division is impossible (Corollary of the Finite Ordinals). The impossible partition was made possible by a denumerable collection of points considered as a complete totality. And since the only property of the collection of points used to define that impossible partition was its supposed denumerability, it was concluded that denumerable collections of points, and in general denumerable sets are inconsistent objects when considered as complete totalities, which is the way they are considered under the hypothesis of the actual infinite. P347 A proof independent of P236-P241 about the inconsistency of the ω-segmentations of any real interval (a, b) will now be given. Let hxi i be an ω-ordered sequence of points in the real interval (a, b), which defines an ω-segmentation of (a, b): (a, x1 ](x1 , x2 ](x2 , x3 ](x3 , x4 ] . . . (5) If the point x1 is removed from hxi i, the remaining points continue to define an ω-segmentation of (a, b): (a, x2 ](x2 , x3 ](x3 , x4 ](x4 , x5 ] . . . (6) If the point x2 is also removed from hxi i, the remaining points continue to define an ω-segmentation of (a, b): (a, x3 ](x3 , x4 ](x4 , x5 ](x5 , x6 ] . . . (7) If the point x3 is also removed from hxi i the remaining points continue to define an ω-segmentation of (a, b): (a, x4 ](x4 , x5 ](x5 , x6 ](x6 , x7 ] . . . (8) It is immediate to prove that for any natural number v it is possible to remove from hxi i the first v elements of hxi i so that the remaining points of hxi i continue to define an ω-segmentation of (a, b). It has been just proved that this is what happens when the first element x1 is removed from hxi i. Suppose that, with n being any natural number, this is also what happens when the first n elements of hxi i are removed from hxi i. We will have the ω-segmentation of (a, b): (a, xn+1 ](xn+1 , xn+2 ](xn+2 , xn+3 ](xn+3 , xn+4 ] . . . (9) Therefore, if the (n + 1)th term is also removed from hxi i, we will also have an ω-segmentation of (a, b): (a, xn+2 ](xn+2 , xn+3 ](xn+3 , xn+4 ](xn+4 , xn+5 ] . . . (10) So, for every natural number v it is possible to remove from hxi i its first v elements, and the remaining elements continue to define an ω-segmentation of (a, b). P348 Suppose that all points that can be removed from hxi i while the remaining ones define an ωsegmentation of (a, b), are removed from hxi i (Principle of Execution). With respect to the number of non-removed points, we will have the following two exhaustive and mutually exclusive alternatives: a) p1: All elements of hxi i are removed from hxi i. b) p2: Not all elements of hxi i are removed from hxi i. Consider c) also the proposition: d) p3: At least one element xs of hxi i was not removed from hxi i. It is clear that p2 ⇒ p3, because if not all elements of hxi i have been removed from hxi i, at least one element xs of hxi i has not been removed from hxi i. But this is impossible because s is a natural number Dividing intervals of space and time 3 and, according to the inductive argument P347, for all natural numbers s it is possible to remove from hxi i its first s elements, and the remaining ones define an ω-segmentation of (a, b). So, it holds: p2 ⇒ p3 ¬p3 ———— ∴ ¬p2 (11) (12) (13) Hence, the proposition p2 is false, and p1 must be true. Consequently, all elements of hxi i can be removed and still have an ω-segmentation of (a, b) P349 The problem is that if all points are removed from the sequence hxi i, the result can only be an empty set of points. This absurdity is a consequence of the actual infinity hypothesis, according to which all points of hxi i exist as a complete and ω-asymmetric totality, whose elements can be considered successively and one by one. The hypothesis of the potential infinity does not lead to the above absurdity, because from this perspective (a, b) can only be divided into a finite number of parts, which can be increased by adding new points, but always having a finite number of parts. P350 It has just been proved that it is possible to remove all points from the sequence of points that defines an ω-segmentation of any real interval (a, b) and still have an ω-segmentation of (a, b). But if all points of a sequence of points are removed from the sequence, the resulting set can only be the empty set. So the absurdity that the empty set of points defines an ω-segmentation of any real interval (a, b) has just been demonstrated. Which allows us to prove the following: Theorem a) P350, of the Divisibility.-The division of a real interval into an infinite number of parts is inconsistent. Proof.-According to P348, ω-segmentations are inconsistent. Since, according to P82, every ω ∗ ordered sequence hx∗i∗ i defines the ω-ordered sequence hxi i, ω ∗ -segmentations are also inconsistent. And since every α-segmentation whose ordinal α is greater than ω contains an ω-segmentation (Theorem of the ωth Term), every α-segmentation is inconsistent. In addition, any non-denumerable segmentation would include infinitely many numerable segmentations, all of them inconsistent. Therefore, the division of a real interval into a numerable or non-numerable infinite number of parts is inconsistent.  Dividing intervals of space and time P351 Supertask theory will be used in this section to confirm Theorem P350. As is well known, space in physics and geometry, and time in physics, are constructions based on the continuum of the real numbers. Let, then, (a, b) be any space interval and (ta , tb ) any time interval, both open, finite and parts of R. Let also hxi i and hti i be two ω-ordered sequences, the first one of points within the interval (a, b) and the second of instants in the interval (ta , tb ), being b the limit of the sequence hxi i, and tb the limit of the sequence hti i. P352 According to the hypothesis of the actual infinity subsumed into the Axiom of Infinity, the infinitely many elements of the sequence hxi i exist all at once, as a complete totality And the same applies to the infinitely many elements of hti i. We can, then, consider one by one the successive elements of hxi i and of hti i. And on that consideration will be based the argument that follows. Indeed, let us consider the following procedure P352: Procedure a) P352. At each of the successive instants of hti i mark each of the successive points of hxi i, so that each point xi is marked at ti , and only at ti . P353 Let us now prove the following two theorems Towards a discrete theory of space and time 4 Theorem a) P353a At instant tb all instants of hti i have passed and all points of hxi i have been marked. Proof.-Being tb the limit of the sequence hti i, the instant tb is the first instant after all instants of the sequence hti i. Therefore, at tb all instants of hti i have passed. On the other hand, the one to one correspondence f between hxi i and hti i defined by xi = f (ti ) proves that at tb all points of hxi i have been marked.  Theorem b) P353b At instant tb not all instants of hti i have passed, and not all points of hxi i have been marked Proof.-Let T be the set of all instants within the interval (ta , tb ) at which only a finite number of instants of the sequence hti i have passed. And let T be the complement set of T with respect to (ta , tb ). It must hold: T = ∅, otherwise there would be at least one t in T and then in (ta , tb ) at which an infinite number of instants of hti i have passed, which is impossible because being tb the limit of hti i, it holds: ∀t ∈ (ta , tb ), ∃v ∈ N : tv < t < tv+1 (14) so that at instant t only a finite number v of instants of the sequence hti i have passed. In consequence, and being t any instant of (ta , tb ), the set of instants of the interval (ta , tb ) at which an infinite number of instants of the sequence hti i have passed is the empty set. And since tb is the first instant after all instants of the interval (ta , tb ), at the instant tb not all instants of the sequence hti i have passed, nor all points of the sequence hxi i have been marked.  P354 The contradiction between the theorems P353a and P353b confirms the Theorem P350 on the inconsistency of dividing a real interval into a denumerable infinitude of parts. Since the continuum of the real numbers is the usual model for space and time, we can generalize the above conclusions in the form of the following: Theorem a) P354 A finite interval of space, or time, cannot be divided into a numerable or non-numerable infinite number pf parts. And if it is not possible to divide a finite interval of space, or of time, into an infinite number of parts, it seems reasonable to consider the possibility of the existence of indivisible minimal units of space and time. Towards a discrete theory of space and time P355 The concepts of point, line, straight line, plane, and angle (and a few more) remain primitive concepts in contemporary geometries, whether Euclidean or non-Euclidean. Although for the last three of them formally productive definitions can be given [8]. Contemporary geometries are also continuous geometries: between any two points of any line* there is always the same number of points: 2ℵo (the power of the continuum). The same number of points that also exist in any two-dimensional surface and in any three-dimensional solid. A line* of one trillionth of a millimeter, for example, has the same number of points as the whole known three-dimensional universe, exactly 2ℵo points. P356 Although they are continuous geometries, all objects in Euclidean and non-Euclidean geometries are made up of points, which are indivisible units. From this perspective, these geometries could be considered as discrete, discontinuous. The problem is that points, whatever they are (if they are anything at all), have no extension. Length is a property of lines, not of points. When two points are joined by a line, the length emerges as a property of the line. Lines can have very different lengths, from ultramicroscopic to intergalactic, although they all have the same number of points. Thus, it could be inferred that the lengths of lines have nothing to do with the number of their corresponding points, although lines have only points, and points, as such points, do not have intrinsic properties. Points only have relative positions in arbitrary references frames. Towards a discrete theory of space and time 5 P357 Being made up of 2ℵo points, any line* contains an uncountable infinity of ω-ordered sequences of points, each of which defines a ω-segmentation in the line* or in any interval of the line. According to the Theorem P354, all of them are inconsistent. Consequently, lines, as objects formed by a continuum of points, are inconsistent objects. Since every two-dimensional surface and every three-dimensional solid is made up of the same uncountable infinitude of points, exactly 2ℵo points, they all are inconsistent objects. And for the same reason, it can be said that the Euclidean space R3 , as defined by a three-dimensional continuum of points, is also inconsistent. A conclusion that is confirmed by the contradictions analyzed in Chapter 12 in relation to the existence of uncountable partitions in the n-dimensional spaces and in the real line. P358 In this chapter and the previous one, it has been proved that: a) A line with two endpoints can only be divided into a finite number of parts with the same finite length. b) The (Euclidean) distance between two points is always finite. c) The length of a (closed or open) line* is always finite. d) lines* of infinite length are inconsistent. e) The division of a finite interval of space (time) into an infinite number of parts of a decreasing length (duration) is inconsistent. f) lines*, as a continuum of points, are inconsistent. g) Two-dimensional and three-dimensional continuums of points are inconsistent. Therefore, it seems reasonable to propose the consideration of a discrete geometry in substitution of the geometries based on continuums of points. Although discrete geometries already exist, they exist for particular purposes, for example the combinatorial analysis of the relationships between geometric elements [3], or the development of computational algorithms for the representation of geometric objects [5, 4]. There are even general discrete geometries, whether or not related to quantum gravity [2, 6, 7, 9], but not independent of infinitist mathematics. This chapter points to a discrete geometry that has nothing to do with the existing discrete geometries, and that it will surely require the development of a discrete and finitist mathematics, free from the inconsistencies caused by the hypothesis of the actual infinity. The discrete and finite nature of space and time will surely bring about an unprecedented revolution in mathematics and physics (see Appendices 37 and 38). P359 In certain discrete geometries, such as the geometries of CALMs (see Appendix 37), the hypotenuse of the right triangles has the same number of indivisible units of space (geons) as the largest of the legs. The factor that converts between discrete hypotenuses and continuous hypotenuses has the same form as the relativistic factor γ of Lorentz transformation. The special theory of relativity could then be interpreted in terms of a discrete geometry, and the interpretation would be compatible with the experimental support of special relativity, a theory of the space-time continuum. Furthermore, the oddities of relativity could be explained and simplified in the new framework of a discrete geometry. Chapter References [1] Aristóteles, Fı́sica, Gredos, Madrid, 1998. [2] Jean Paul Van Bendegem, In defense of discrete space and time, Logique et Analyse 38 (1997), 150 –152. [3] Károly Bezdek, Classical topics in discrete geometry, Springer., New York, 2010. [4] Li M. Chen, Digital and discrete geometry, Springer, New York, 2014. [5] Satyan L. Devadoss and Joseph O’Rourke, Discrete and computational geometry, Princeton University Press, Princeton and Oxford, 2011. [6] Fabrizio Fiore, Luciano Burderi, Tiziana Di Salvo, Marco Feroci, Claudio Labanti, Michelle R. Lavagna, and Simone Pirrotta, HERMES: a swarm of nano-satellites for high energy astrophysics and fundamental physics, Space Telescopes and Instrumentation 2018: Ultraviolet to Gamma Ray (Jan-Willem A. den Herder, Shouleh Nikzad, and Kazuhiro Nakazawa, eds.), vol. 10699, International Society for Optics and Photonics, SPIE, 2018. [7] Laurent Freidel and Etera R. Livine, Bubble networks: framed discrete geometry for quantum gravity, General Relativity and Gravitation 51 (2018), no. 1, 9. [8] Antonio Leon, Proving unproved euclidean propositions on a new foundational basis, International Journal of Scientific Research in Mathematical and Statistical Sciences 7 (2020), no. 3, 61–81. [9] Alexander I. Nesterov and Héctor Mata, How nonassociative geometry describes a discrete spacetime, Frontiers in Physics 7 (2019), no. 32, 1–18. 6