Eur. Phys. J. C (2019) 79:937 https://doi.org/10.1140/epjc/s10052-019-7456-2
Regular Article - Theoretical Physics
Well-order as a construction principle for physical theories
Peter Schusta Sebastian-Bauer-Str. 37a, 81737 Munich, Germany
Received: 19 August 2019 / Accepted: 4 November 2019 / Published online: 18 November 2019 © The Author(s) 2019
Abstract Physics has up to now missed to express in math- together with a binary order relation R that is itself a subset ematical terms the fundamental idea of events of a path in of the set of all ordered pairs A × A of A;see[1, p 10, p 17], time and space uniquely succeeding one another. An appro- [2, p 64, p 80]. priate mathematical concept that reflects this idea is a well- A densely ordered set is an ordered set in which between ordered set. In such a set every subset has a least element. any two elements of the set exists another element of the set; Thus every element of a well-ordered set has as its definite see [2, p 205]. Well known examples for this are the rational successor the least element of the subset of all elements larger numbers or the real numbers together with the natural order than itself. This is apparently contradictory to the densely of numbers; see [2, p 213]. By employing the real number ordered real number lines which conventionally constitute line as coordinate axes this is transferred into the resulting the coordinate axes in any representation of time and space Rn. The open balls as the base of the Euclidean topology are and in which between any two numbers exists always another a reflection of this. For any radius r > 0 exists a non-empty number. In this article it is shown how decomposing this dis- open ball around any x ∈ Rn. Since Riemannian manifolds accord in favour of well-ordered sets causes spacetime to be are locally homeomorphic to a suitable Rn the dense order discontinuous. of R is not only a characteristic of Newtonian and Special relativistic spacetime but also of the description of space and time in General Relativity.1 1 Introduction Another type of ordered sets are well-ordered set in which every non-empty subset has a least element; see [2, p 224]. Whenever the motion of a particle is described in classical, Therefore every element of a well-ordered set has as its defi- special relativistic or general relativistic space and time it is nite successor the least element of the subset of all elements always implicitly assumed that the particle moves from an larger than itself. This evidently provides appropriate means initial event to a final event through a sequence of consecu- to mathematically represent the idea of events in spacetime tive intermediate events. In other words, it is presumed that uniquely succeeding one another. The defining example for every element of the path has an unique successor. But even well-ordered sets are ordinal numbers. Among these the nat- though this is a fundamental concept, it is never disclosed. ural numbers together with the natural order of numbers are It is neither named in Newton’s laws of motion or his law of the best known instance. But they are by far not the only gravitation nor mentioned in the basic principles of Special members of this class of numbers. In fact, they just open the or General Relativity. Consequently, it is never expressed in gates into this transfinite realm. mathematical terms. Also without further ado these theories make constitu- tively use of the real number line by employing it as (local) 1 n coordinate axes or coordinate axes and parameter line for In Riemannian geometry R is more than a mere vehicle to locally handle a manifold. As stated right at the top of [3, p 1] a smooth mani- their respective manifolds which represent space and time. fold is resembling Euclidean space enough to permit the establishment Unfortunately these very two unnamed axioms are con- of all essential features of calculus. This statement is realised in the fol- tradictory to each other. This becomes apparent most clearly lowing by defining local coordinate systems to be homeomorphisms, n in terms of ordered sets. An ordered set (A, R) is a set A i.e. smooth manifolds and R are topologically equivalent. By this the standard topology of Rn, whose base are open balls, and hence the dense order of R, on which calculus hinges, is fundamentally introduced into a e-mail: [email protected] smooth manifolds. 123 937 Page 2 of 11 Eur. Phys. J. C (2019) 79 :937
Densely ordered sets and well-ordered sets are obviously not fully part of the main line of reasoning of this article. But irreconcilable. A familiar example for this incongruity is the it is nevertheless worth noting because it describes how the countability of Q: bijections form Q to N are known to exist need to keep the disjoint ordered and unordered subsets of a but none of them preserve the order of Q. partial causal order apart during linear transformations leads Since each of the two unnamed axioms of spacetime to c. respectively incorporates one of these types of ordered sets they cannot be assumed in combination. 2 Causality and order types As it is shown in [4]or[5, chapter 1.6] the concept of ordered sets is already a part of the definition of Minkowski As mentioned in the introduction, the mathematical represen- space. However, with regard to the question if Minkowski tation of the idea of “a distinct next neighbour” is the con- ( ,
On the other hand, every ordinal number is well-ordered be associated to a x0 ∈ (R,<). Only the x0-coordinates are and for every ordinal number α ={β ∈ Ord | β<α<ω1} restrained to grow to larger values in their respective orders. less than the smallest uncountable ordinal number ω1 exists In case of a well-order (7), Lorentz transformations are an order embedding into (Q,<), i.e. every countable ordi- understood to be causal automorphismen of the embedding nal number α<ω1 can be represented by a monotonic space sequence of rational numbers; see [2, p 214, theorem 3] M ⊃ M(α) =×3 (Ri (α), <), (8) or [6, p 350/351]. By an equally monotonic injection this i=0 sequence is easily transferred into (R,<). Due to the fact which also lends its quadratic form to its subspaces. that every element of α ={β ∈ Ord | β<α<ω1} Requiring the exclusive choice between (6) xor (7)tobe has a unique successor there is always a non-empty inter- conserved by Lorentz transformations seems at first glance val ∅ = (q(β), q(β + 1)) ⊂ Q between their images which to enlarge the set of invariants of the causal automorphismen is transferred by an injection to ∅ = (x(β), x(β + 1)) ⊂ R. group of M. But this requirement is already a member of Therefore, the set that represents α in Q or R is a discontin- said set because, as has just been shown, the causality con- uous union of singletons: dition (2) actually states the formation of an ordered type, or respectively, an ordered set (A, R ⊂ A × A). Firstly, the Q(α) = {q(β)} constraint of events being put in sequence along the x0-axis β<α by the standard order < determines the binary relation R: ={q ∈ Q|q ≥ q(0)}− (q(β), q(β + 1)) R =<. (9) β<α Secondly, the choice of a set (4) → R(α) = {x(β)} A = R A = R(α) (10) β<α of whose set of ordered pairs < is a subset ={x ∈ R|x ≥ x(0)}− (x(β), x(β + 1)) . <⊂ R × R <⊂ (α) × (α) β<α R R (11) (5) completes the assembly of an order type, or respectively, an ordered set: For N α<ωthe first sets in (4) and (5) reduce to ( , ⊂ × ) = (R,<⊂ R × R) [q(0), q(α)] and [x(0), x(α)]. A R A A In summary, if events are to have distinct successors and if this order is to be expressed by the natural order <,they (A, R ⊂ A × A) = (R(α), <⊂ R(α) × R(α)) . (12) have to be elements of a countable ordinal number α<ω1 to which (R,<)merely serves as an embedding space. Notably, As the contradiction in the term “infinitesimal next neigh- admissible events necessarily form a discontinuous subset bour” reveals, physics cannot avoid to encompass the full of (R,<). Quantisation is the special case of discontinuity extent of the causality condition (2). The use of a well-order where events are evenly timed and spaced. the author is biased to, may be declared by replacing M by (α) By Lorentz transformations the spatial dimensions con- M in (1); cf the rhs of (8). tribute to all other admissible time axes. For this reason, they For a sequence of events on the cone the above argumenta- must be ordered by the same order type as the time axis or tion is valid if it is restricted to sequences of events for which otherwise the order type chosen for the x0-coordinate differs the relation x y between different frames of reference: x0 < y0 (13) Q(c; y − x) =−c2(y0 − x0)2 + (y − x)2 = 0(14) x0 ∈ (R,<) ⇒ x j ∈ (R,<) , j = 1, 2, 3(6) is transitive, i.e. for straight lines on the cone. For more details of the prevalence of time as the carrier 0 ∈ 0(α), < ⇒ j ∈ i (α), < , = , , . x R x R j 1 2 3 of causal order over the spatial dimensions as it is expressed (7) by ⇒ in (6) and (7) see Appendix A. A question that immediately arises when utilising well- The Ri (α), i = 0, 1, 2, 3 are different representations of α, ordered spacetime is how to handle an how to interpret space- which clearly exist. Though the R j (α), j = 1, 2, 3 are well- times that are based on ordinal numbers containing limit ordi- ordered this does not say their order has to be followed strictly nals larger than ω =|N|. To this end M(α) is not really apt. A in every step. As long as (3) is satisfied their respective values mathematical framework that is more suitable for these pur- 0 j can be associated to a xβ as freely as the x of (R,<) can poses are the surreal numbers which are discussed in Sect. 3. 123 937 Page 4 of 11 Eur. Phys. J. C (2019) 79 :937
( 0, 1, 2, 3) 3 Surreal spacetime (17a). Hereby, the event x0 x0 x0 x0 is furnished with a −yβ i surrounding of subfinite numbers ω rβ ei . The coefficient − An appropriate mathematical ground for investigating well- ω yβ>0 = 1 of any of those subfinite orders of magnitude ωyβ>0 ordered spacetimes is provided by Conway’s surreal numbers can be turned into an event in spacetime if (17a) is multi- No [7]. This class of numbers comprises real numbers as well plied by the corresponding inverse infinite order of magni- ω 1 β> as infinite and subfinite elements like and ω .In[7] elements tude ωy 0 : like 1 are called infinitesimal elements. To avoid confusion ω ωyδ = ωyβ i + ω−yδ+yδ i with anything that is based on the concept of infinitesimal s rβ ei x0ei 1 β<δ i=0,1,2,3 vicinity (see second paragraph of Sect. 2), ω ∈ No and alike = , , , i 0 12 3 elements are called subfinites in this article. For the concep- − + ω yβ ri e , (18a) tual reasons for this distinction see Appendix B. β i 1 δ<β<α Since No is also a field, ω is the multiplicative inverse of i=0,1,2,3 ω in the exact sense of the group operation No × No → No i = i . 1 −1 x0 rδ (18b) representing multiplication on No: ω ω = ω ω = 1 ∈ N ⊂ R ⊂ No. Moreover, every surreal number is uniquely The three terms in (18a) sustainably spark a perception of expandable in terms of orders of magnitude of ω, called its past (β<δ↔ ωyβ ω0 ↔ yβ<δ > 0), present (β = δ ↔ − + − normal form: ω yδ yδ = ω0 = 1) and future (β>δ↔ ω yβ ω0 ↔ yβ>δ < 0).2 Thus the exponents of the orders of magnitude = ωyβ , s rβ (15a) establish a causal order of events, which is the task already β<α committed to the x0-components of events. These two orders α, β : , ordinal numbers (15b) are accordable by an order-reversing bijection between the 0 yβ : sequence of surreal numbers decreasing along α, sets {yβ } and {rβ }:
(15c) 0 0 f :{yβ }↔{−rβ } ,β∈ α, yβ ∈ No, rβ ∈ R. (19) rβ ∈ R\0. (15d) By this the inverted well-ordering (15c) of the sequence {yβ } Members of No4 may then be written as becomes relevant to the set {r 0}. Conversely, {r 0}∈R β β restricts α to countable ordinal numbers as there are no = ωyβ 0 + 1 + 2 + 3 s rβ e0 rβ e1 rβ e2 rβ e3 uncountable well-ordered subsets of (R,<)[1], [6, p 350]. β<α For violation of causality occurring when a well-order and a yβ i = ω rβ ei . (16) dense order are collocated see Sect. 4. β<α Summarised, in (17a) and (18a) α is strictly less than ω1 i=0,1,2,3 0 0 and rβ ∈ (R (α), <). In yet other words: when the inverted i 4 Other than in (15) there are a suitable number ofrβ allowed well-order of the order of magnitudes of an element s of No to be 0 so that the sums in all components si of the form (15) in its normal form is harmonised with the natural order of real range over the same order of magnitudes. This lets emerge a numbers, No4 relegates to its subset of elements of countable 4 α = ( )<ω complete copy of R with basis {ei } at each order of magni- normal length nl s 1 with causally ordered mem- tude ωyβ from which the coefficient for the associated order bers of different ωyβ R4 as candidates for events in spacetime: of magnitude can be selected. The first step in making (16) S4 (α) = useful for physics is to designate the zeroth order of mag- causal⎧ ⎫ ⎪ ⎪ nitude ω0 = 1 to be position space, or more precisely, to ⎨⎪ ⎬⎪ − 0 designate the coefficients ri of ω0 to be events in spacetime = ω rβ i |α<ω ∧ κ<ρ: 0 < 0 0 ⎪s rβ ei 1 rκ rρ ⎪ i = i ≡ i ⎩⎪ β<α<ω ⎭⎪ r0 x0 x : 1 i=0,1,2,3 = ω0 i + ω−yβ i , 4 s x0ei rβ ei (17a) ⊂ No . (20) i=0,1,2,3 0<β<α = , , , ∈ S4 (α) i 0 1 2 3 In order to establish the speed of light in each s causal − 0 i = i ≡ i , = , , , , rβ r0 x0 x i 0 1 2 3 (17b) it must be subject to the further restraint that each ω rβ = − 0 − 0 = > − . rβ 0 1 2 3 rβ 4 y0 0 yβ>0 (17c) ω (rβ , rβ , rβ , rβ ) ∈ ω R is a member of the cone < > Additionally, in (17c) y0 = 0 is chosen to be the largest Cβ (x0) = Cβ (x0) ∪ Cβ (x0) ∪ x0, (21a) order of magnitude so by virtue of (15c) all other orders of magnitude yβ>0 have to be negative. This is reflected by the 2 ωa ωb says that ωb is of a higher order of magnitude than ωa;see according negative sign in the exponents in the second sum of [8, p 54]. 123 Eur. Phys. J. C (2019) 79 :937 Page 5 of 11 937 C <( ) = ∈ R4|Q( ; − ) ≤ ∧ 0 < 0 , β x0 rβ c rβ x0 0 rβ x0 step-by-step instead of being regarded as an already complete entity which deterministically fixes a path. (21b) Reversely, a stage-like M(α) is transformed into M(α) by C >( ) = ∈ R4|Q( ; − ) ≤ ∧ 0 > 0 , β x0 rβ c rβ x0 0 rβ x0 firstly generating the set of all path (2) and (3) allow in (8) and ×3 ( i (α), <) (21c) repeating this step for all combinations i=0 R of all order embeddings of α. After this, aligning all these paths where (21b) correlates to the first term in (18a), (21c)to along orders of magnitude by using the negative of the r 0- ω M(α) the third term in (18a) and x0 is the second term in (18a) components of their points as exponents for leads to . denominating the present event in spacetime. For fixed κ<δ When employing well-ordered Minkowski space by means and δ<ρ(21b) and (21c) have to be iteratively true: of path-inclined causal normal forms of surreal numbers (23) acknowledging the question how to handle spacetimes < < < rκ ∈ Cκ (rκ ) ⊂···⊂Cβ (rβ ) ⊂···⊂Cδ (rδ) involving limit ordinals λ, which is pending since the end of λ ⇔ ∈ C <( ) = C κ<δ( ), Sect. 2, should no longer be postponed. Limit ordinals are rκ β rβ rκ (22a) α ω κ<β≤δ elements of every ordinal number larger than and are > > > characterised by lacking a definite predecessor. Hence the rρ ∈ Cρ (rρ) ⊂···⊂Cβ (rβ ) ⊂···⊂Cδ (rδ) 0 corresponding (rλ, rλ) is approached but cannot be reached > ρ>δ 0 ⇔ rρ ∈ Cβ (rβ ) = C (rρ). (22b) by stepping out the trail of (rβ<λ, rβ<λ). The full and math- δ≤β<ρ ematically exact usability of the realms of the subfinite and the infinite within the framework of surreal numbers facili- α All in all, for fixed the set of candidates for events in space- tates the following answer to this question: Only coefficients time is composed of the following surreal numbers: 0 of orders of magnitude whose exponent rλ corresponds to a λ ∈ α M(α) ={ ∈ S4 (α)|κ<δ: ∈ C κ<δ( ) limit ordinal are permitted to become reality: s causal rκ rκ ρ>δ ∧ρ>δ: rρ ∈ C (rρ)}. (23) δ ∈{λ ∈ α>ω|λ : limit ordinal} (24)
0 −r0 (20)Ð(22) are the analogues of (2) and (3) resp. (A.4a) and Multiplying a s ∈ M(α > ω) by ωrλ sends all ω δ<β<λrδ<β<λ 4 (A.4b) for subsets of No with the addition of harmonising between the present event ω0rδ and its designated successor 0 0 the inverse well-order of the orders of magnitude with the 0 (−rδ<β<λ+rλ ) ωrλ rλ directly to ω rδ<β<λ, i.e. directly from the order of position space to make (23) fully internally ordered. future (−r 0 < 0) to the past (−r 0 + r 0)>0 with- 4 δ<β<λ δ<β<λ λ It should be clear by now that No is utilised in terms of out a stop at the present. These sections of virtual events paths. Every element of (23) constitutes a discontinuous path ∈ M(α) 0 of s (23) therefore offer stages on which addi- ωrδ to be paced out by multiplication by eligibly selected .The tional structures may play out any desired real feature. “Real” information about the transition from r0 to rδ is contained in 0 denotes at first real numbers promptly implying accessibility ( + rδ ,ωrδ ) the pair r0 0 ; for a detailed discussion of this with ωrδ to measurement. Examples for such virtual formation pro- regard to topology see Appendix B. Obviously forces act cesses of real features that immediately come to mind are whenever the rβ do not form a straight line and/or the distance the generation of a dressed inertial mass from bare iner- between consecutive coefficients rβ+1 − rβ and rβ+2 − rβ+1 tial masses or the shaping of a dressed electric charge from varies. Furthermore every rδ is a point of intersection for bare electric charges. Producing divergences while constitut- many paths each of which offers a different succeeding event. ing the final real feature along a sequence of virtual stages In this vein M(α) (23) acts as a stage of potential next events. should be avoidable by using suitably chosen subfinite bare But even that there are ω1-many paths intersecting at rδ, instances of the desired feature on each virtual stage. Two R4 i.e. that there is a full 1-cone of successors rπ available of the measures for the suitability of the subfiniteness of the at rδ, this does not cure the contradiction in the statement bare instances of the feature under consideration are surely “infinitesimal next neighbour”, because each of this potential the length of a virtual section and the weight of a single vir- next events rπ can only be reached by means of a discontin- tual stage in it. If multiple features are to be formed between uous path. And what is even more important, all potential two real events the index set for real events (24) needs to be events with time component less than that of the successor refined to limit ordinals which themselves comprise a suf- 0 0 0 rη which became reality rδ < rπ < rη cannot be revisited, ficient number of limit ordinals to furnish enough virtual as it would be necessary to establish the density needed for structure to assemble all that is requested. In such situations an infinitesimal vicinity, or else causality is breached. See mass may serve as a bracket with inertial mass being the first Sect. 4 for more on this. On the other hand the interpreta- feature to arise and gravitational mass being the last charac- tion of every rπ having a probability to be picked as the next teristic to be accrued until its numerical value is that of inertial real event is readily at hand. A s ∈ M(α) is then established mass. GTR strongly suggests that the subfinite bare instances 123 937 Page 6 of 11 Eur. Phys. J. C (2019) 79 :937 of every other feature are accompanied by a subfinite bare This necessity to choose is highly visible in every member instance of gravitational mass, unless the bare instances of of the set the other features are the bare instances of gravitational mass Sn(( (R)) ⊇ ω ) = themselves, merely in disguise. Thereby the well-order of ⎧ ord 1 ⎫ spacetime transfers to all other physical theories as a guid- ⎪ ⎪ ⎨⎪ ⎬⎪ ing principle; for some effects of this on path integration = ω−χ i (χ) | (χ) ∈×n (R,<) ⎪s r ei r i=0 ⎪ see Sect. 5. By the respective orientation and distances of ⎩⎪ (R) ⎭⎪ ∈ M(α) χ∈(R, − exp (tv) = γv(t), t t 0 Δt = t + − t = . k 1 k + (31) t ∈ R. (29) N 1 By reason that the order type of spacetime is not dense, it is This shows indeed that every point of γv ⊂ M between o and not possible to turn Δt into an infinitesimal dt. Accordingly p is accessible by means of expo. But the idea of smoothly for the associated sum the transition into an integral is barred, progressing from o to p, i.e. visiting all elements of γv in even though Δt is assumed to be already small enough to their natural order by consecutively transiting to a distinct permit H(Q(tk), P(tk−1)) → H(qk, pk−1). Now this first next neighbour is not included in (29). If it were a part of simple deduction only states a restriction. But starting out (29), the real interval [0, 1]⊂R in its entirety would have to from the definition of the Riemann integral, there is another be the representation of an (necessarily innumerable) ordinal way to obtain the same result which casts some light on the number which it is not. Since Riemannian geometry is fun- full ordinal quality of N: damentally build on the concept of infinitesimal vicinities it is not very surprising that the notion of elements of tangent b n f (x)d(x) := lim f (ξ )(x − x − ), (32a) spaces link elements of a Riemannian manifold as definite →∞ i i i 1 a n next neighbours that (28) hints at does not resonate with the i=1 n rest of the theory. The ignorance of members of a interval of n P ([a, b] ,ξ) = xi , xi−1 , (32b) γv this requires is for example in contradiction with the topo- i=1 logical concept of neighbourhoods which are constitutively a = x0 <ξ1 < x1 < ···< xn <ξn < xn = b. (32c) woven into the idea of a manifold; cf [3, p 71, proposition 1 30] and footnote 1. The limit on the rhs of (32a) is understood in terms of For discontinuously charting a geodesic (or any other infinitesimal refinement of the tagged partition (32b), that Sn(( (R)) ⊇ ω ) curve) (27) is the instrument of choice. In ord 1 is to say, every ξ ∈[a, b] is approximated from the left and γ surely exists a s which produces v when successively mul- the right respectively by a sequence {xn }. Essentially, this ωχ = i(n) tiplied by all ; see also Appendix C. For n 4 all sub- procedure is the same as the one by which R is constructed sets of this s corresponding to a countable ordinal number from Q. In consequence, a single map N →[a, b]⊂Q does and whose coefficients of the involved orders of magnitudes not suffice for the intended purpose. Instead, the whole set additionally satisfy (23) are eligible for physics. As only the NQ([a, b]) ={g|g : N →[a, b]⊂Q} of functions from coefficients of orders of magnitude associated with a limit N to [a, b]⊂Q is needed. Since already 2N = N{0, 1}↔ ordinal may become reality (24), the coefficients of all other N {q1, q2}, q1,2 ∈ Q is of the same cardinality as c, the cardi- orders of magnitude, i.e. the virtual events, do not need to nality of NQ([a, b]) ⊃ N{q , q } must at least be c;infact,it γ 1 2 adhere to v. Thus all members of (27) with physical subsets is the same. Thus, in (32a) the vague ∞ should be replaced whose real events are congruent, are affiliated to γv from the by an ordinal number γ ≥ ω1, or, more precisely, the limit angle of physics. should be omitted and the sum should run from 1 to γ . The approximation of a function f by simple functions fn supports this claim: 5 Well-ordered spacetime and path integration k − 1 k In,k = , , (33a) 2n 2n Some first implications of selecting a well-ordered spacetime 2n+ 2n+ for quantum theory are attainable by just jumping right in the 21 k − 1 21 k − 1 fn = 1A , = 1 −1( ), (33b) middle of path integration. For this purpose, the most useful 2n n k 2n f In,k part of this method is the step in the derivation of the orthodox k=1 k=1 form of the bosonic path integral just before introducing time k = 1, 2,...,22n + 1, (33c) integration in the exponential function (see for example [9]): f = fN = lim fn. (33d) n→∞ N N+1 + 1 If for a non-constant continuous function f :[a, b]→R q ; t |q; t= dqk,a dpk− ,a 1 2π the objective of the procedure (33) really is to realise every k=1 a k=1 a value of f by a single indicator function, the limit must aim N+1 at least at ω1. Otherwise, there will be not enough terms in × exp i (qk,a − qk−1,a)pk−1,a the sum (33b) to exhaust the range of f completely. In unison k=1 a with this, 2n and 22n should be read as 2N and 22N to indicate N+1 × exp −i H(q , p − )Δt , (30) the necessary cardinality of the process (33). Replacing them k k 1 γ ≥ ω k=1 by an ordinal number 1 converts the rhs of (33b)into 123 937 Page 8 of 11 Eur. Phys. J. C (2019) 79 :937 4 ∈ M ((R,<))⊂{×4 ( i : R → R)}=R(R4), the transfinite expression that it is at its heart. See [7] for an qk,a i=0 g 1 axiomatic definition of terms like γ . (37) As it was shown in Sect. 2, a well-ordered spacetime R R →|M ((R,<))|≤| (R4)|=| R|=f, (38) consists of representations of a countable ordinal number i α<ω1. Hence, if (30) is evaluated over such a space- where the g are restricted by versions of (21)Ð(22) adapted time, N has to be strictly less than ω1 and the associated to a dense order and a real spacetime; the order embedding in sum in the exponential function cannot turn into an inte- (20) turns into confining g0 to be strictly monotonic increas- i 0 gral. For ω This results in a full R3 of simultaneous events at every x0 , = , , Appendix A: Minkowski space as an ordered set spanned by the e j j 1 2 3. In the sense that the members of the subspaces x0 × R3 defy to be ordered by their x0- In order to rethink what role the requirements (2) and (3) component, simultaneity is a concept antagonistic to causal- fulfil and which weight they carry in establishing causality ity. On a more positive note it is justifiable to regard the mem- and the formation of Minkowski space it is useful to start out bers of each of these subspaces of simultaneity to be equal: ( 0, ) ∈ R3 0 > 0 from a plain R4 and (re)assemble M step by step. From the point of view of x x every y at y x ( 0, ) At first, all coordinate axes of R4 are equal which is syn- is a suitable unordered element of a successor y y .(Yes, ∈ R3 onymous to stating that there is no preferred direction in R4. every y ; c is not yet established, wait for it.) This idea More formally this is to say a plain R4 is understood to be an of equality is easily extended to incorporate transformations × × 6 unordered set (R4, ∅). Any order in a sequence made of its of coordinates in the space(s) spanned by e1 e2 e3 since 0 elements has to be recorded by an external parameter. This the order constituted by the x -component is not impaired by approach to organise members of R4 clearly leans too much them. This shows that equipping the task to describe an order towards Newton to be of fundamental use for physics. But it solely to one dimension preserves in terms of the dimension- highlights the fact that causality is an independent requisition ality of subspaces as much as possible of the equality of R4 demanding its own mathematical structure to carry it. If the directions of a plain . At the same time it underscores 0 order of a sequence of elements of R4 is to be read off the from a different angle that x as the carrier of causality is elements themselves the mathematical structure to carry this discerned from the other dimensions. R1+3 causality has to be sought among the four coordinate axes With causality now an immanent trait of the set con- <1+3⊂ R4 × R4 of R4. One way of making causality an inherent feature of stituting the order must be the same in R1+3 R4 in this manner is to endow all four axes with the task to all of ’s admissible coordinate systems. That is to say, <1+3 0 display the order of a sequence of elements of R4: members of must have different x -coordinates respec- tively in all permissible coordinate systems. Clearly <1+3 < ⇔ i < i x y x y for all i concurrently. (A.1) excludes transformations which mingle the e0-axis with one , , , = , , By this realisation of inherent causality conceptual unifor- or more of e1 e2 e3 because any e j j 1 2 3 resulting mity of all four axes is maintained. But it is so overly redun- from this is a direction along which elements are unordered in 0 dant that no periodicity can occur. Using on the other hand contradiction to all of them having different x -coordinates respectively, i.e. being ordered, in the unprimed frame of 5 Since limN→∞ is used in [10] in the usual way, i.e. just unspecificly aiming at larger values and thus to ω, and moreover it is claimed that 6 There is one subspace of simultaneity at the origin stating to which (41) exhibits a continuous result already after this most initial subset of events it is unrelated. This subspace is readily transferred to other ele- ω1, it may be assumed that (41) is an expression which is structurally ments by translation. Talking subsequently about a single unordered as plain as the natural numbers; see footnote 4. subspace refers to this blueprint at the origin. 123 937 Page 10 of 11 Eur. Phys. J. C (2019) 79 :937 reference. A way to overcome this stringent separation is to The above line of argumentation is valid for both kinds preclude elements from <1+3 so that coordinate axes of the of spacetime, densely ordered or well-ordered. Just as the now enlarged unordered subspace which include the current R j , j = 1, 2, 3in(6) are not forced to grow by their under- j e0-direction are rendered possible. This is exactly what (3) lying dense order the R (α) in (7) are not forced to take on does: every event strictly outside the cone is deemed causally larger values by their underlying well-order. Only R0 and unordered: R0(α) are coerced to do so. But, as explained, both spatial parts are bound to the order type of their respective time axes x <4 y ⇔ x0 < y0 ∧ Q(y − x) ≤ 0, (A.4a) as to not interfere with the creation of other time directions, 1 i.e. with the designation of other directions in the ordered , ∈/<4 ⇔ 0 = 0 ∨ Q( − )> . x y 1 x y y x 0 (A.4b) part of M or M(α) as the carrier of the order. In addition, using a double cone to separate causally ordered events from causally unordered events ensures the linearity of the coordinate transformations which preserve the estab- Appendix B: Comparing surreal numbers to topology lished order; see [4]. With the coordinate axis carrying the causal order considered as time and the space spanned by Other than to avoid confusion there are also conceptual rea- the remaining axes to be the spatial dimensions the concept sons for linguistically distinguishing the subfinite elements 1 of a maximum velocity c ensues. The numerical value of c of No like ω from the notion of infinitesimal vicinities of indicates how many formerly ordered events with x0 = y0 standard calculus. Topologies are closed (only) under finite were excluded from <1+3. intersections of their open sets. The very root of this axiom is Summarised, (2) expresses the fact that a causal order is to avoid producing inevitably open sets of singletons, i.e. to n implemented into R4 by singling out only one of the coor- allow for instance for R other topologies than the discrete dinate axes as its carrier thereby maintaining as much of topology. As a result of this, the notion of convergence resp. n the equality of all coordinate axes of a plain R4 as possi- infinitesimal vicinity in R with the ball topology is commit- n ble. The resulting R1+3 does not admit any order preserv- ted to distances between elements of R which despite ever ing coordinate transformation that mix the e0 direction bear- reducing nonetheless never leave the real numbers. Infinites- 1+3 ing < and the unordered directions e1, e2, e3. A well- imal vicinities like these are the basis for ever improvable visible hallmark of this is the lack of suitable candidates inward bound approximations. { 1 | ∈ N} for new axes spanning the unordered subspace of simul- Using in their right sets all elements of sets like 2n n 1+3 taneity among the directions in R involving e0.Thisis Conway’s surreal numbers do on a regular basis from day healed by (3) which firstly diminishes <1+3 enough to let ω on what topology axiomatically circumvents for the real { 1 | ∈ N} order isomorphic coordinate transformations incorporating numbers. By bouncing all elements of 2n n offof0, all coordinate axes be allowed. Secondly, its double cone structure refines these transformation to be linear and leads 1 1 1 1 = 0| ..., , , , 1 (B.5) to the emergence of an invariant maximum velocity. Thus ω 8 4 2 = R4 = (R4,<4) <4 the formation of M 1 1 is completed; 1 is defined by (A.4). The invariant division of causally ordered is created which opens the gates to the realm of subfinite ele- from causally unordered events furnished by (3) is reflected ments. Consequently there are non-singleton subsets of No4 by the Lorentz transformations in that these do not permit the which contain only one real element r ∈ Rn and one or more R4 0 time axis to transit into the unordered region of 1 outside 1 n subfinite elements r0 + r s , r ∈ R , s ∈ No. Multiplying the cone and reversely prohibit the spatial axes to enter the ω one of those surrounding subfinite elements by its inverse ordered region inside the cone. s s infinite order of magnitude ω and substracting r0ω gives The idea of events being causally unordered xor causally the real element r. In this sense the subfinites surrounding r0 ordered can be concised as follows: Each coordinate axis in (in cooperation with the infinite elements) provide a nexus R4 is ordered by the natural order of numbers. The equality of connecting r0 to other real numbers which prevents r0 from a number of these coordinate axes in R4 is synonymous to the being isolated. In contrast to infinitesimal approximation this absence of a preferred direction among them which in turn is operation is inherently bound outward. tantamount to be able to choose freely values for the associ- If causality is not an issue the nexus can of course be a ated coordinates. Every subspace of R4 for which this holds dense set (or have any other desired characteristic). With true is regarded as causally unordered. A causally ordered axis is distinguished from this by restricting the choice of 1 values for its associated coordinate to values greater than the s((Q, 123