THREE DIFFERENT FORMALISATIONS OF EINSTEIN’S RELATIVITY

THE REVIEW OF SYMBOLIC LOGIC Page 1 of 19

THREE DIFFERENT FORMALISATIONS OF EINSTEINrsquoSRELATIVITY PRINCIPLE

JUDIT X MADARAacuteSZ GERGELY SZEacuteKELY

Alfreacuted Reacutenyi Institute of Mathematics Hungarian Academy of Sciencesand

MIKE STANNETT

Department of Computer Science The University of Sheffield

Abstract We present three natural but distinct formalisations of Einsteinrsquos special principleof relativity and demonstrate the relationships between them In particular we prove that they arelogically distinct but that they can be made equivalent by introducing a small number of additionalintuitively acceptable axioms

sect1 Introduction The special principle of relativity (SPR) which states that the lawsof physics should be the same in all inertial frames has been foundational to physicalthinking since the time of Galileo and gained renewed prominence as Einsteinrsquos firstpostulate of relativity theory (Einstein 1916 sect1)

If a system of coordinates K is chosen so that in relation to it physicallaws hold good in their simplest form the same laws hold good in rela-tion to any other system of coordinates Krsquo moving in uniform translationrelatively to K This postulate we call the ldquospecial principle of relativityrdquo

Despite its foundational status the special principle of relativity remains problematicdue to its inherent ambiguity (Szaboacute 2004 Goumlmoumlri amp Szaboacute 2013ab) What after all dowe mean by ldquophysical lawsrdquo and what does it mean to say that the ldquosame lawsrdquo hold intwo different frames

These ambiguities often lead to misunderstandings and misinterpretations of the princi-ple See eg Muller (1992) for the resolution of one such misinterpretation We believethat formalisation is the best way to eliminate these ambiguities In this paper we investi-gate the principle of relativity in an axiomatic framework of mathematical logic Howeverwe will introduce not one but three different naturally arising versions of the principle ofrelativity not counting the parameters on which they depend such as the formal languageof the framework used

It is not so surprising when one tries to capture SPR formally that more than oneldquonaturalrdquo version offers itself ndash not only was Einsteinrsquos description of his principle givenonly informally but its roots reach back to Galileorsquos even less formal ldquoship storyrdquo (Galileo1953 pp 186ndash187)

Received May 22 20162010 Mathematics Subject Classification 03B30 83A05Key words and phrases special principle of relativity special relativity relativity theory axiomatic

method first order logic

ccopy Association for Symbolic Logic 2017

1 doi101017S1755020317000065

available at httpswwwcambridgeorgcoreterms httpsdoiorg101017S1755020317000065Downloaded from httpswwwcambridgeorgcore University of Sheffield Library on 06 Apr 2017 at 123513 subject to the Cambridge Core terms of use

2 JUDIT X MADARAacuteSZ ET AL

Since all three of the versions we investigate are ldquonaturalrdquo and simply reflect differentapproaches to capturing the original idea there is no point trying to decide which is theldquoauthenticrdquo formalisation The best thing we can do is to investigate how the differentformalisations are related to each other Therefore in this paper we investigate under whichassumptions these formalisations become equivalent

In a different framework but with similar motivations Goumlmoumlri and Szaboacute also intro-duce several formalisations of Einsteinrsquos ideas (Szaboacute 2004 Goumlmoumlri amp Szaboacute 2013abGoumlmoumlri 2015) Intuitively what they refer to as ldquocovariancerdquo corresponds to our principlesof relativity and what they call the principle of relativity is an even stronger assumptionHowever justifying this intuition is beyond the scope of this paper as it would require usto develop a joint framework in which both approaches can faithfully be interpreted

11 Contribution In this paper we present three logical interpretations (SPRMSPR+ and SPRBIOb) of the relativity principle and investigate the extent to which theyare equivalent We find that the three formalisations are logically distinct although theycan be rendered equivalent by the introduction of additional axioms We prove rigorouslythe following relationships

111 Counter-examples and implications requiring no additional axioms (Fig 1)

bull SPR+ rArr SPRM (Theorem 71)bull SPRBIOb rArr SPRM (Theorem 71)bull SPRBIOb rArr SPR+ (Theorem 71)bull SPRM rArr SPR+ (Theorem 72)bull SPRM rArr SPRBIOb (Theorem 73)

112 Adding axioms to make the different formalisations equivalent (Fig 2)

bull SPR+ rArr SPRBIOb assuming AxId AxEv AxIB AxField (Theorem 74)bull SPRBIOb rArr SPRM assuming L = L0 AxEv AxExt (Theorem 77)bull SPR+ rArr SPRM assuming L = L0 AxId AxIB AxField AxEv AxExt (The-

orems 74 77)bull SPRBIOb rArr SPR+ assuming L = L0 AxEv AxExt (Theorems 72 77)

Fig 1 Counter-examples and implications requiring no additional axioms


THREE FORMALISATIONS OF EINSTEINrsquoS RELATIVITY PRINCIPLE 3

Fig 2 Axioms required to make the different formalisations equivalent

113 SPRM SPR+ and the decomposition of SPRBIOb into SPRIOb and SPRB

(Fig 3)

bull SPR+ rArr SPRIOb assuming AxId AxEv (Theorem 75)bull SPR+ rArr SPRB assuming AxIB AxField (Theorem 76)

Outline of the paper We begin in sect2 by characterising what we mean by a ldquolaw of naturerdquoin our first-order logic framework Rather than going into all the difficulties of definingwhat a law of nature is we focus instead on the requirement that all inertial observersmust agree as to the outcomes of experimental scenarios described by such a law In sect3we give some examples in sect4 we demonstrate our three formalisations of SPR in sect5 wediscuss the types of models our language admits and in sect6 we state the axioms that willbe relevant to our results These results are stated formally in sect7 In sect8 we discuss somealternative assumptions to axiom AxIB The proofs of the theorems can be found in sect9

Fig 3 SPRM SPR+ and the decomposition of SPRBIOb into SPRIOb and SPRB



We conclude with a discussion of our findings in sect10 where we also highlight questionsrequiring further investigation

sect2 Laws of nature Before turning our attention towards formalising the principleof relativity we need to present the framework in which our logical formalisms will beexpressed Following the approach described in (Andreacuteka Madaraacutesz amp Neacutemeti 2007Andreacuteka Madaraacutesz Neacutemeti amp Szeacutekely 2011) we will use the first-order logical (FOL)3-sorted language

L0 = IObBQ 0 1+ middotWas a core language for kinematics In this language

bull IOb is the sort of inertial observers (for labeling coordinate systems)bull B is the sort of bodies ie things that movebull Q is the sort of quantities ie numbers with constants 0 and 1 addition (+) and

multiplication (middot)bull W is the worldview relation a 6-ary relation of type IOb times B times Q4

The statement W(k b p) represents the idea that ldquoinertial observer k coordinatisesbody b to be at spacetime location prdquo

Throughout this paper we use h k and their variants to represent inertial observers(variables of sort IOb) we use b and c to represent bodies (variables of sort B) and pq and r are variables of type Q4 The sorts of other variables will be clear from context

Given this foundation various derived notions can be defined

bull The event observed by k isin IOb as occurring at p isin Q4 is the set of bodies that kcoordinatises to be at p

evk(p) equiv b isin B W(k b p)bull For each k h isin IOb the worldview transformation wkh is a binary relation on

Q4 which captures the idea that h coordinatises at q isin Q4 the same event that kcoordinatises at p isin Q4

wkh(p q) equiv [evk(p) = evh(q)] (wdef)

bull The worldline of b isin B as observed by k isin IOb is the set of locations p isin Q4 atwhich k coordinatises b

wlinek(b) equiv p W(k b p)We try to choose our primitive notions as simple and ldquoobservationally orientedrdquo as

possible cf Friedman (1983 p 31) Therefore the set of events is not primitive but rathera defined concept ie an event is a set of bodies that an observer observes at a certain pointof its coordinate system Motivation for such a definition of event goes back to Einstein andcan be found in Misner et al (1973 p 6) and Einstein (1996 p 153)

Since laws of nature stand or fall according to the outcomes of physical experiments wenext consider statements φ which describe experimental claims For example φ might sayldquoif this equipment has some specified configuration today then it will have some expectednew configuration tomorrowrdquo This is very much a dynamic process-oriented description ofexperimentation but since we are using the language of spacetime the entire experimentcan be described as a static four-dimensional configuration of matter in time and spaceWe therefore introduce the concept of scenarios ie sentences describing both the initialconditions and the outcomes of experiments Although our scenarios are primarily intended



to capture experimental configurations and outcomes they can also describe more complexsituations as illustrated by the examples in sect3 One of our formalisations of SPR will bethe assertion that all inertial observers agree as to whether or not certain situations arerealizable Our definition of scenarios is motivated by the desire to have a suitably large setof sentences describing these situations

To introduce scenarios formally let us fix a language L containing our core languageL0 We will say that a formula

φ equiv φ(k x) equiv φ(k x1 x2 x3 )

of language L describes a scenario provided it has a single free variable k of sort IOb(to allow us to evaluate the scenario for different observers) and none of sort B Theother free variables x1 can be thought of as experimental parameters allowing usto express such statements as φ(k v) equiv ldquok can see some body b moving with speed vrdquoNotice that numerical variables (in this case v) can sensibly be included as free variableshere but bodies cannot ndash if we allow the use of specific individuals (Thomas say) wecan obtain formulae (ldquok can see Thomas moving with speed vrdquo) which manifestly violateSPR since we cannot expect all observers k to agree on such an assertion The truth valuesof certain formulas containing bodies as free variables can happen to be independent ofinertial observers for example ν2 in sect3 but we prefer to treat these as exceptional cases tobe proven from the principle of relativity and the rest of the axioms

Thus φ equiv φ(k x) represents a scenario provided

bull k is free in φ(k x)bull k is the only free variable of sort IObbull the free variables xi are of sort Q (or any other sort of L representing mathematical

objects) andbull there is no free variable of sort B (or any other sort of L representing physical

objects)

The set of all scenarios will be denoted by ScenariosFinally for any formula φ(k x) with free variables k of sort IOb and x1 x2 of any

sorts the formula

AllAgree〈φ〉 equiv (forallk h isin IOb)((forallx)[φ(k x) harr φ(h x)]

)

captures the idea that for every evaluation of the free variables x all inertial observersagree on the truth value of φ Let us note that AllAgree〈〉 is defined not just for scenarioseg it is defined for the nonscenario examples of sect3 too

In sect4 one of the formalisations of SPR will be that AllAgree〈φ〉 holds for every possiblescenario φ

sect3 Examples Here we give examples for both scenarios and nonscenarios To be ableto show interesting examples beyond the core language used in this paper let us expandour language with a unary relation Ph of light signals (photons) of type B and a functionM B rarr Q for rest mass ie M(b) is the rest mass of body b For illustrative purposeswe focus in particular on inertial bodies ie bodies moving with uniform linear motionand introduce the notations speedk(b) = v and velk(b) = (v1 v2 v3) to indicate that bis an inertial body moving with speed v isin Q and velocity (v1 v2 v3) isin Q3 accordingto inertial observer k These notions can be easily defined assuming AxField introducedin sect6 and their definitions can be found eg in Andreacuteka Madaraacutesz Neacutemeti amp Szeacutekely(2008) Madaraacutesz Stannett amp Szeacutekely (2014)



In the informal explanations of the examples below we freely use such modal expres-sions as ldquocan set downrdquo and ldquocan send outrdquo in place of ldquocoordinatiserdquo to make theexperimental idea behind scenarios intuitively clearer and to illustrate how the dynamicalaspects of making experiments are captured in our static framework See also (Molnaacuter ampSzeacutekely 2015) for a framework where the distinction between actual and potential bodiesis elaborated within first-order modal logic

Examples for scenarios

bull Inertial observer k can set down a body at spacetime location (0 0 0 0)

φ1(k) equiv (existb)W(k b 0 0 0 0)

bull Inertial observer k can send out an inertial body with speed v

φ2(k v) equiv (existb)speedk(b) = v

bull Inertial observer k can send out an inertial body at location (x1 x2 x3 x4) withvelocity (v1 v2 v3) and rest mass m

φ3(k x1 x2 x3 x4 v1 v2 v3m) equiv(existb)[W (k b x1 x2 x3 x4) and velk(b) = (v1 v2 v3) and M(b) = m]

bull The speed of every light signal is v according to inertial observer k

φ4(k v) equiv (forallb)[Ph(b) rarr speedk(b) = v]

Let us consider scenario φ4 to illustrate that AllAgree〈φ〉 means that all inertial obser-vers agree on the truth value of φ for every evaluation of the free variables x Assumethat the speed of light is 1 for every inertial observer ie that (forallk)(forallb)[Ph(b) rarrspeedk(b) = 1] and (existb)Ph(b) holds Then the truth value of φ4(k 1) is true for everyinertial observer k but the truth value of φ4(k a) is false for every inertial observer k ifa = 1 Thus AllAgree〈φ4〉 holds

Examples for nonscenarios

bull The speed of inertial body b according to inertial observer k is v

ν1(k v b) equiv speedk(b) = v

Then AllAgree〈ν1〉 means that all inertial observers agree on the speed of each bodyObviously we do not want such statements to hold

Notice incidentally that it is possible for all observers to agree as to the truth value ofa nonscenario but this is generally something we need to prove rather than assert a prioriFor example consider the nonscenario

bull The speed of light signal b is v according to inertial observer k

ν2(k v b) equiv Ph(b) rarr speedk(b) = v

Then AllAgree〈ν2〉 means that all inertial observers agree on the speed of each lightsignal and it happens to follow from AllAgree〈φ4〉 where scenario φ4 is given aboveTherefore AllAgree〈ν2〉 will follow from our formalisations of SPR which entail the truthof formula AllAgree〈φ4〉

While different observers agree on the speed of any given photon in special relativitytheory they do not agree as to its direction of motion which is captured by the followingnonscenario



bull The velocity of light signal b is (v1 v2 v3) according to inertial observer k

ν3(k v1 v2 v3 b) equiv Ph(b) rarr velk(b) = (v1 v2 v3)

Then AllAgree〈ν3〉 means that all inertial observers agree on the velocity of each lightsignal and once again we do not want such a formula to hold

sect4 Three formalisations of SPR

41 First formalisation A natural interpretation of the special principle is to identifya set S of scenarios on which all inertial observers should agree (ie those scenarios weconsider to be experimentally relevant) If we now define

SPR(S) equiv AllAgree〈φ〉 φ isin S (1S)

the principle of relativity becomes the statement that every formula in SPR(S) holds Forexample if we assume that all inertial observers agree on all scenarios and define

SPR+ equiv AllAgree〈φ〉 φ isin Scenarios (1+)

then we get a ldquostrongest possiblerdquo version of SPR(S) formulated in the language LIt is important to note that the power of SPR(S) (and hence that of SPR+) strongly

depends on which language L we use It matters for example whether we can only use L toexpress scenarios related to kinematics or whether we can also discuss particle dynamicselectrodynamics etc The more expressive L is the stronger the corresponding principlebecomes

42 Second formalisation A natural indirect approach is to assume that the world-views of any two inertial observers are identical In other words given any model M ofour language L and given any observers k and h we can find an automorphism of themodel which maps k to h while leaving all quantities (and elements of all the other sortsof L representing mathematical objects) fixed That is if the only sort of L representingmathematical objects is Q we require the statement

SPRM equiv (forallk h isin IOb)(existα isin Aut (M))[α(k) = h and αQ = I dQ] (2)

to hold where I dQ is the identity function on Q and αQ denotes the restriction of α tothe quantity part of the model If L has other sorts representing mathematical objects thanQ then in (2) we also require αU = I dU to hold for any such sort U

43 Third formalisation Another way to characterise the special principle of relativ-ity is to assume that all inertial observers agree as to how they stand in relation to bodiesand each other (see Fig 4) In other words we require the formulae

SPRB equiv (forallk kprime)(forallb)(existbprime)[wlinek(b) = wlinekprime(bprime)] (3B)

and

SPRIOb equiv (forallk kprime)(forallh)(existhprime)[wkh = wkprimehprime] (3IOb)

to be satisfied We will use the following notation

SPRBIOb = SPRBSPRIObSPRBIOb is only a ldquotiny kinematic slicerdquo of SPR It says that two inertial observers are

indistinguishable by possible world lines and by their relation to other observers



Fig 4 Illustration for SPRB and SPRIOb

sect5 Models satisfying SPR According to Theorems 72 and 73 below SPRM is thestrongest of our three formalisations of the relativity principle since it implies the othertwo without any need for further assumptions

The lsquostandardrsquo model of special relativity satisfies SPRM and therefore also SPR+ andSPRBIOb where by the lsquostandardrsquo model we mean a model determined up to isomor-phisms by the following properties (a) the structure of quantities is isomorphic to that ofreal numbers (b) all the worldview transformations are Poincareacute transformations (c) forevery inertial observer k and Poincareacute transformation P there is another observer h suchthat wkh = P (d) bodies can move exactly on the smooth timelike and lightlike curvesand (e) worldlines uniquely determine bodies and worldviews uniquely determine inertialobservers

In fact there are several models satisfying SPRM in the literature and these modelsalso ensure that the axioms used in this paper are mutually consistent Indeed in (Szeacutekely2013 Andreacuteka Madaraacutesz Neacutemeti Stannett amp Szeacutekely 2014 Madaraacutesz amp Szeacutekely 2014)we have demonstrated several extensions of the lsquostandardrsquo model of special relativity whichsatisfy SPRM Applying the methods used in those papers it is not difficult to showthat SPRM is also consistent with classical kinematics Again this is not surprising asthere are several papers in the literature showing that certain formalisations of the prin-ciple of relativity cannot distinguish between classical and relativistic kinematics and asIgnatowski (1910 1911) has shown when taken together with other assumptions SPRimplies that the group of transformations between inertial observers can only be thePoincareacute group or the inhomogeneous Galilean group For further developments of thistheme see (Leacutevy-Leblond 1976 Borisov 1978 Pal 2003 Pelissetto amp Testa 2015)

The simplest way to get to special relativity from SPRM is to extend the languageL0 with light signals and assume Einsteinrsquos light postulate ie light signals move withthe same speed in every direction with respect to at least one inertial observer Then bySPRM AxPh follows ie light signals move with the same speed in every directionaccording to every inertial observer AxPh even without any principle of relativity implies(using only some trivial auxiliary assumptions such as AxEv (see p 9)) that the transfor-mations between inertial observers are Poincareacute transformations see eg (Andreacuteka et al2011 Theorem 22)

It is worth noting that SPRM also admits models which extend the lsquostandardrsquo model ofspecial relativity for example models containing faster-than-light bodies which can interact



dynamically with one another (Szeacutekely 2013 Andreacuteka et al 2014 Madaraacutesz amp Szeacutekely2014)

sect6 Axioms We now define various auxiliary axioms As we show below whether ornot two formalisations of SPR are equivalent depends to some extent on which of theseaxioms one considers to be valid

In these axioms the spacetime origin is the point e0 = (0 0 0 0) and the unit pointsalong each axis are defined by e1 = (1 0 0 0) e2 = (0 1 0 0) e3 = (0 0 1 0) ande4 = (0 0 0 1) We call e0 e4 the principal locations

AxEv All observers agree as to what can be observed If k can observe an event some-where then h must also be able to observe that event somewhere

(forallk h)(forallp)(existq)[evk(p) = evh(q)]

AxId If k and h agree as to whatrsquos happening at each of the 5 principal locations thenthey agree as to whatrsquos happening everywhere (see Fig 5)

(forallk h)((foralli isin 0 4)[evk(ei ) = evh(ei )] rarr (forallp)[evk(p) = evh(p)]

)

We can think of this axiom as a generalised form of the assertion that all worldviewtransformations are affine transformations

AxExtIOb If two inertial observers coordinatise exactly the same events at every possiblelocation they are actually the same observer

(forallk h)((forallp)[evk(p) = evh(p)] rarr k = h

)

AxExtB If two bodies have the same worldline (as observed by any observer k) then theyare actually the same body

(forallk)(forallb bprime)[wlinek(b) = wlinek(bprime) rarr b = bprime]

AxExt We write this as shorthand for AxExtB and AxExtIObAxField (Q 0 1+ middot) satisfies the most fundamental properties of R ie it is a field

(in the sense of abstract algebra see eg (Stewart 2009))

Notice that we do not assume a priori that Q is the field R of real numbers because wedo not know and cannot determine experimentally whether the structure of quantities in thereal world is isomorphic to that of R Moreover using arbitrary fields makes our findingsmore general

AxIB All bodies (considered) are inertial ie their worldlines are straight lines accordingto every inertial observer

(forallk)(forallb)(existp q)[q = e0 and wlinek(b) = p + λq λ isin Q]

Fig 5 Schematic representation of AxId If k and h agree as to whatrsquos happening at each of the5 principal locations then they agree as to whatrsquos happening at every location p



AxIB is a strong assumption In sect8 we introduce generalisations of AxIB allowingaccelerated bodies too We choose to include AxIB in our main theorems because itis arguably the simplest and clearest of these generalisations The main generalisationwlDefM of AxIB is a meta-assumption and the others are quite technical assertions whichare easier to understand in relation to AxIB

sect7 Results If M is some model for a FOL language L and is some collection oflogical formulae in that language we write M to mean that every σ isin is validwhen interpreted within M If 1 2 are both collections of formulae we write 1 2to mean that

M 2 whenever M 1

holds for every model M of L For a general introduction to logical models see(Mendelson 2015 Marker 2002)

Theorem 71 demonstrates that our three formalisations of the principle of relativity arelogically distinct It is worth noting that based on the ideas used in the proof of Theorem 71it is also easy to construct sophisticated counterexamples to their equivalence extending thelsquostandardrsquo model of special relativity

THEOREM 71 The formalisations SPRM SPR+ and SPRBIOb are logically distinct

bull M SPR+ rArr SPRMbull M SPRBIOb rArr SPRMbull M SPRBIOb rArr M SPR+

By Theorems 72 and 73 SPRM is the strongest version of the three formalisationssince it implies the other two without any extra assumptions

THEOREM 72 SPRM rArr M SPR+

THEOREM 73 SPRM rArr M SPRBIOb

Theorem 74 tells us that SPR+ can be made as powerful as SPRBIOb by addingadditional axioms This is an immediate consequence of Theorems 75 and 76

THEOREM 74 SPR+ cup AxIdAxEvAxIBAxField SPRBIOb

THEOREM 75 There exist scenarios ψ ψ such that SPR(ψ ψ) cup AxIdAxEv SPRIOb

THEOREM 76 There exists a scenario ξ such that SPR(ξ) cup AxIBAxField SPRB

Theorem 77 tells us that equipping SPRBIOb with additional axioms allows us torecapture the power of SPRM (and hence by Theorem 72 SPR+)

THEOREM 77 Assume L = L0 Then M SPRBIOb cup AxEvAxExt rArr SPRM

Thus although SPRM SPRBIOb and SPR+ are logically distinct they become equiv-alent in the presence of suitable auxiliary axioms

sect8 Alternatives to AxIB In this section we generalise AxIB to allow discussion ofaccelerated bodies

For every model M we formulate a property which says that world lines are paramet-rically definable subsets of Q4 where the parameters can be chosen only from Q For thedefinitions cf (Marker 2002 sect116 sect121)



wlDefM For any k isin IOb and b isin B there is a formula ϕ(y1 y2 y3 y4 x1 xn)where all the free variables y1 y2 y3 y4 x1 xn of ϕ are of sort Q and there isa isin Qn such that wlinek(b) equiv q isin Q4 M ϕ(q a)We note that plenty of curves in Q4 are definable in the sense above eg curves which

can be defined by polynomial functions as well as the worldlines of uniformly acceleratedbodies in both special relativity and Newtonian kinematics

In general not every accelerated worldline is definable ndash indeed the set of curves whichare definable depends both on the language and the model For example uniform circularmotion is undefinable in many models however if we extend the language with the sinefunction as a primitive notion and assert its basic properties by including the appropriateaxioms then uniform circular motion becomes definable

By Theorems 81 and 82 assumptions AxIB and AxField can be replaced by wlDefMin Theorem 74 (Theorem 82 follows immediately from Theorems 75 and 81)

THEOREM 81 (wlDefM and M SPR+) rArr M SPRB

THEOREM 82 (wlDefM and M SPR+ cup AxIdAxEv) rArr M SPRBIOb

We note that M AxIBAxField rArr wlDefM Moreover wlDefM is more generalthan AxIB assuming AxField The disadvantage of wlDefM is that it is not an axiom buta property of model M We now introduce for every natural number n an axiom AxWl(n)which is more general than AxIB assuming AxField and n ge 3 and stronger than wlDefM

AxWl(n) Worldlines are determined by n distinct locations ie if two worldlines agreeat n distinct locations then they coincide

(forallk kprime)(forallb bprime)[(exist distinct p1 pn isin wlinek(b) cap wlinekprime(bprime)) rarr wlinek(b) = wlinekprime(bprime)]

We note that AxIBAxField AxWl(2) and AxWl(n) AxWl(i) if i ge nFurthermore for every n we have M AxWl(n) rArr wlDefM To see that this must

be true assume M AxWl(n) choose k isin IOb and b isin B let p1 pn isin wlinek(b) bedistinct locations and define

ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn

4 ) equiv(existh)(existc)[(y1 y2 y3 y4) (x

11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]

Then it is easy to see that wlinek(b) equiv q isin Q4 M ϕ(q p1 pn) whencewlDefM holds as claimed

By Theorems 83 and 84 Theorems 74 and 76 remain true if we replace AxIB andAxField with AxWl(n)

THEOREM 83 SPR+ cup AxIdAxEvAxWl(n) SPRBIOb

THEOREM 84 There is a scenario ξ such that SPR(ξ) cup AxWl(n) SPRB

sect9 Proofs We begin by proving a simple lemma which allows us to identify when twoobservers are in fact the same observer This lemma will prove useful in several places below

LEMMA 91 AxEvAxExtIOb (forallk h hprime)[(wkh = wkhprime) rarr h = hprime]

Proof Suppose wkh = wkhprime and choose any p isin Q4 Let q isin Q4 satisfy evk(q) =evh(p) so that wkh(q p) holds (q exists by AxEv) Since wkh = wkhprime it follows thatwkhprime(q p) also holds so that evh(p) = evk(q) = evhprime(p) This shows that h and



hprime see the same events at every p isin Q4 whence it follows by AxExtIOb that h = hprimeas claimed

Proof of Theorem 71 SPR+ rArr SPRM SPRBIOb rArr SPRM SPRBIOb rArrSPR+

Constructing the required counterexamples in detail from scratch would be too lengthyand technical and would obscure the key ideas explaining why such models exist Accord-ingly here we give only the lsquorecipesrsquo on how to construct these models

To prove that SPR+ does not imply SPRM let Mminus be any model satisfying SPRMminusand AxExt and containing at least two inertial observers and a body b such that the world-lines of b are distinct according to the two observers Such models exist see eg the onesconstructed in Szeacutekely (2013) The use of AxExt ensures that distinct bodies have distinctworldlines Let us now construct an extension M of Mminus (violating AxExt) by addinguncountably-infinite many copies of body b as well as countably-infinite many copies ofevery other body Clearly M does not satisfy SPRM since this would require the exis-tence of an automorphism taking a body having uncountably many copies to one havingonly countably many copies Nonetheless M satisfies SPR+ since it can be elementarilyextended to an even larger model M+ satisfying SPRM+ (and hence by Theorem 72SPR+) by increasing the population of other bodies so that every body has an equal(uncountable) number of copies (see (Madaraacutesz 2002 Theorem 2820)) Thus M sat-isfies SPR+ but not SPRM In more detail Let M+ be an extension of M obtainedby increasing the population of bodies so that every body has an equal (uncountable)number of copies We will use the TarskindashVaught test (Marker 2002 Prop 235 p 45) toshow that M+ is an elementary extension of M Let φ(vw1 wn) be a formula andsuppose a1 an in M and d+ in M+ satisfy M+ φ(d+ a1 an) We have tofind a d in M such that M+ φ(d a1 an) If d+ is not a body then d+ is alreadyin M since we extended M only by bodies Assume then that d+ is a body Then d+has infinitely many copies in M so we can choose d a copy of d+ in M such thatd isin a1 an Let α be any automorphism of M+ which interchanges d and d+ andleaves every other element fixed Then α(a1) = a1 α(an) = an and α(d+) = dBy M+ φ(d+ a1 an) we have that M+ φ(α(d+) α(a1) α(an)) ThusM+ φ(d a1 an) as required

To prove that SPRBIOb does not imply SPRM or SPR+ let M be any model ofSPRBIOb and AxExt containing at least two inertial observers k and h and a body bfor which wlinek(b) = (0 0 0 t) t isin Q = wlineh(b) Such models exist seeeg (Szeacutekely 2013) Duplicating body b leads to a model in which SPRBIOb is stillsatisfied since duplicating a body does not change the possible worldlines but it vio-lates both SPRM (the automorphism taking one inertial observer to another cannot takea body having only one copy to one having two) and SPR+ (since scenario ϕ(m) equiv(forallb c)[wlinem(b) = wlinem(c) = (0 0 0 t) t isin Q rarr b = c] holds for h but does nothold for k)

Proof of Theorem 72 SPRM rArr M SPR+

Suppose SPRM so that for any observers k and h there is an automorphism α isinAut (M) such that α(k) = h and α leaves elements of all sorts of L representing math-ematical objects fixed

We will prove that AllAgree〈φ〉 holds for all φ isin Scenarios ie given any observersk and h any scenario φ and any set x of parameters for φ we have

φ(k x) harr φ(h x) (1)



To prove this choose some α isin Aut (M) which fixes Q and all the other sortsrepresenting mathematical objects and satisfies α(k) = h

Suppose M φ(k x) Since α is an automorphism φ(α(k) α(x)) also holds in MBut α(k) = h and α(x) = x so this says that M φ(h x) Conversely if φ(h x) holdsin M then so does φ(k x) by symmetry

Proof of Theorem 73 SPRM rArr M SPRBIOb

Suppose SPRM Then (forallk h)(existα isin Aut (M))[α(k) = h and αQ = I dQ] We wish toprove that M satisfies

SPRIOb equiv (forallk kprime)(forallh)(existhprime)[wkh = wkprimehprime]

SPRB equiv (forallk kprime)(forallb)(existbprime)[wlinek(b) = wlinekprime(bprime)]

Recall that whenever α is an automorphism of M and R is a defined n-ary relation on Mwe have

R(v1 vn) lArrrArr R(α(v1) α(vn)) (2)

Proof of SPRIOb Choose any k kprime h We need to find hprime such that wkh = wkprimehprime so letα isin Aut (M) be some automorphism taking k to kprime and define hprime = α(h) Now itrsquos enoughto note that wkh(p q) is a defined 10-ary relation on M (one parameter each for k and h4 each for p and q) so that

wkh(p q) lArrrArr wα(k)α(h)(α(p) α(q))

holds for all p q by (2) Substituting α(k) = kprime α(h) = hprime and noting that α leaves allspacetime coordinates fixed (because αQ = I dQ) now gives

wkh(p q) lArrrArr wkprimehprime(p q)

as requiredProof of SPRB Choose any k kprime and b We need to find bprime such that wlinek(b) =

wlinekprime(bprime) As before let α isin Aut (M) be some automorphism taking k to kprime definebprime = α(b) and note that ldquop isin wlinek(b)rdquo is a defined 6-ary relation on M Applying (2)now tells us that

wlinek(b) = wlineα(k)(α(b)) = wlinekprime(bprime)as required

Proof of Theorem 74 SPR+ cup AxIdAxEvAxIBAxField SPRBIOb

This is an immediate consequence of Theorems 75 and 76 Proof of Theorem 75 SPR(ψ ψ) cup AxIdAxEv SPRIOb for some ψ ψ

Given a 5-tuple of locations xi = (x0 x1 x2 x3 x4) let match (see Fig 6) be therelation

match(h k xi ) equiv (foralli isin 0 4)(evh(ei ) = evk(xi ))

and define ψ ψ isin Scenarios by

ψ(k p q xi ) equiv (existh)[(evh(p) = evk(q)) and match(h k xi )]


Choose any k h kprime In order to establish SPRIOb we need to demonstrate some hprime suchthat wkh = wkprimehprime To do this let xi be such that evk(xi ) = evh(ei ) for all i = 0 4(these exist by AxEv)



Fig 6 Schematic showing the behaviour of function match which tells us which locations in krsquosworldview correspond to the principle locations in hrsquos worldview

Then in particular ψ(k e0 x0 xi ) equiv (existh)[(evh(e0) = evk(x0)) and match(h k xi )]holds Since SPR(ψ ψ) it follows that ψ(kprime e0 x0 xi ) also holds so there is some hprimesatisfying

match(hprime kprime xi ) (3)

Now choose any p and q We will show that evh(p) = evk(q) if and only if evhprime(p) =evkprime(q) whence wkh = wkprimehprime as claimed

Case 1 Suppose evh(p) = evk(q) In this case ψ(k p q xi ) holds and we need toprove that evhprime(p) = evkprime(q) It follows from SPR(ψ ψ) that ψ(kprime p q xi ) also holdsie there exists hprimeprime satisfying

(evhprimeprime(p) = evkprime(q)) and match(hprimeprime kprime xi ) (4)

It follows from (3) that

match(hprime kprime xi ) and match(hprimeprime kprime xi )

holds ie

evhprime(ei ) = evkprime(xi ) = evhprimeprime(ei )

holds for all i = 0 4 By AxId it follows that (forallr)(evhprime(r) = evhprimeprime(r))It now follows from (4) that

evhprime(p) = evhprimeprime(p) = evkprime(q)

ie evhprime(p) = evkprime(q) as requiredCase 2 Suppose evh(p) = evk(q) In this case ψ(k p q xi ) holds and we need to

prove that evhprime(p) = evkprime(q) It follows from SPR(ψ ψ) that ψ(kprime p q xi ) also holdsie there exists hprimeprime satisfying


As before it follows from (3) that evhprime(ei ) = evkprime(xi ) = evhprimeprime(ei ) holds for all i =0 4 and hence by AxId that (forallr)(evhprime(r) = evhprimeprime(r))

It now follows from (5) that


ie evhprime(p) = evkprime(q) as required Proof of Theorem 76 SPR(ξ) cup AxIBAxField SPRB for some ξ

We define ξ isin Scenarios by ξ(k p q) equiv (existb)(p q isin wlinek(b))To see that this satisfies the theorem suppose that AxIB and AxField both hold and

choose any k kprime isin IOb and any b isin B We will demonstrate a body bprime satisfyingwlinek(b) = wlinekprime(bprime) whence SPRB holds as claimed



According to AxIB there exist points pk qk isin Q4 where qk = e0 and

wlinek(b) = pk + λqk λ isin QIn particular therefore the points p = pk and q = pk + qk are distinct elements of thestraight line wlinek(b)

This choice of p and q ensures that the statement ξ(k p q) holds By SPR(ξ) it followsthat ξ(kprime p q) also holds ie there is some body bprime such that p q isin wlinekprime(bprime) By AxIBwlinekprime(bprime) is also a straight line

It follows that wlinek(b) and wlinekprime(bprime) are both straight lines containing the same twodistinct points p and q Since there can be at most one such line (by AxField) it followsthat wlinek(b) = wlinekprime(bprime) as claimed

Proof of Theorem 77 If L = L0 then M SPRBIOb cupAxEvAxExt rArr SPRM

Choose any k kprime We need to demonstrate an automorphism α isin Aut (M) which is theidentity on Q and satisfies α(k) = kprime

bull Action of α on IOb Suppose h isin IOb According to SPRIOb there exists some hprimesuch that wkh = wkprimehprime By Lemma 91 this hprime is uniquely defined (since we wouldotherwise have distinct hprime hprimeprime satisfying wkprimehprime = wkprimehprimeprime) Define α(h) = hprime

bull Action of α on B Suppose b isin B According to SPRB there exists some bprime such thatwlinek(b) = wlinekprime(bprime) By AxExtB this bprime is uniquely defined Define α(b) = bprime

bull Action on Q Define αQ = I dQ Notice that this forces α(p) = p for all p isin Q4

We already know that α fixes Q It remains only to prove that α isin Aut (M) and thatα(k) = kprime

Proof that α(k) = kprime Recall first that for any equivalence relation R it is the case thatR = R Rminus1 and that given any k kprime kprimeprime we have

bull wkk is an equivalence relationbull wminus1

kkprime = wkprimek andbull wkkprime wkprimekprimeprime = wkkprimeprime (by AxEv and (wdef))

By construction we have wkk = wkprimeα(k) whence wkprimeα(k) is an equivalence relationIt now follows that

wkprimeα(k) = wkprimeα(k) wminus1kprimeα(k)

= wkprimeα(k) wα(k)kprime

= wkprimekprime

and hence by Lemma 91 that α(k) = kprime as required

Proof that α isin Aut (M) We know that wlinek(b) = wlinekprime(α(b)) = wlineα(k)(α(b))or in other words given any b and q isin Q4

b isin evk(q) lArrrArr α(b) isin evα(k)(q) (6)

We wish to prove W(h b p) harr W(α(h) α(b) p) for all h b and p (recall thatα(p) = p) This is equivalent to proving

b isin evh(p) lArrrArr α(b) isin evα(h)(p) (7)

Choose any h b p Let q isin Q4 satisfy evk(q) = evh(p) ndash such a q exists by AxEv Then

evα(k)(q) = evα(h)(p) (8)



because wkh = wkprimeα(h) = wα(k)α(h) by the definition of α(h) and the fact establishedabove that kprime = α(k)

To prove (7) let us first assume that b isin evh(p) Then b isin evk(q) so α(b) isin evα(k)(q)by (6) Then by (8) we have α(b) isin evα(h)(p) Next choose b isin evh(p) = evk(q) Thenα(b) isin evα(k)(q) = evα(h)(p) by (6) and (8) It follows that b isin evh(p) lArrrArr α(b) isinevα(k)(q) as required

It remains to show that α is a bijection

Proof of injection

bull Observers Suppose α(h) = α(hprime) Then by the definition of α we have

wkh = wkprimeα(h) = wkprimeα(hprime) = wkhprime

and now h = hprime by Lemma 91bull Bodies Suppose α(b) = α(c) By definition of α we have

wlinek(b) = wlinekprime(α(b)) = wlinekprime(α(c)) = wlinek(c)

and now b = c follows by AxExtB

Proof of surjection We need for every hprime bprime that there are h b satisfying α(h) = hprime andα(b) = bprime

bull Observers Let hprime isin IOb By SPRIOb there exists h such that wkprimehprime = wkh andnow hprime = α(h) for any such h

bull Bodies Let bprime isin B By SPRB there exists b isin B such that wlinekprime(bprime) = wlinek(b)and now bprime = α(b) for any such b

This completes the proof Proof of Theorem 81 (wlDefM and M SPR+) rArr M SPRB

Assume M SPR+ and wlDefM Choose any k kprime isin IOb and any b isin B We willdemonstrate a body bprime satisfying wlinek(b) = wlinekprime(bprime)

Let ϕ(y x) equiv ϕ(y1 y2 y3 y4 x1 xn) be a formula such that all the free variablesy x of ϕ are of sort Q and choose a isin Qn such that

wlinek(b) equiv q isin Q4 M ϕ(q a)Such ϕ and a exist by wlDefM

We define ψ isin Scenarios by ψ(h x) equiv (existc)(forallq)[q isin wlineh(c) harr ϕ(q x)]Clearly M ψ(k a) Then by SPR+ M ψ(kprime a) Thus there is bprime isin B such thatwlinekprime(bprime) equiv q isin Q4 M ϕ(q a) and wlinek(b) = wlinekprime(bprime) for this bprime

Proof of Theorem 82 (wlDefM and M SPR+cupAxIdAxEv) rArr M SPRBIOb

This is an immediate consequence of Theorems 75 and 81 Proof of Theorem 83 SPR+ cup AxIdAxEvAxWl(n) SPRBIOb

This is an immediate consequence of Theorems 75 and 84 Proof of Theorem 84 SPR(ξ) cup AxWl(n) SPRB for some ξ

We define ξ isin Scenarios by ξ(k p1 pn) equiv (existb)(p1 pn isin wlinek(b)) It iseasy to check that ξ satisfies the theorem and we omit the details



sect10 Discussion and conclusions In this paper we have shown formally that adoptingdifferent viewpoints can lead to different but equally lsquonaturalrsquo formalisations of the spe-cial principle of relativity The idea that different formalisations exist is of course notnew but the advantage of our approach is that we can investigate the formal relationshipsbetween different formalisations and deduce the conditions under which equivalence canbe restored

We have shown in particular that the model-based interpretation of the principleSPRM is strictly stronger than the alternatives SPR+ and SPRBIOb and have identifiedvarious counterexamples to show that the three approaches are not in general equivalentOn the other hand equivalence is restored in the presence of various axioms We notehowever that the following question remains open since it is unclear whether SPR+ isenough in its own right to entail SPRBIOb

CONJECTURE 101 SPR+ rArr SPRB

An interesting direction for future research would be to investigate the extent to whichour existing results can be strengthened by removing auxiliary axioms For example ourproof that SPRM can be recovered from SPR+ currently relies on L = L0 AxId AxIBAxField AxEv and AxExt While we know that some additional axiom(s) must berequired (since we have presented a counterexample showing that SPR+ rArr SPRM)the question remains whether we can develop a proof that works over any language L orwhether the constraint L = L0 is required Again assuming we allow the same auxiliaryaxioms how far can we minimise the set S of scenarios while still entailing the equivalencebetween SPR(S) and SPRM

sect11 Acknowledgement The authors would like to thank the anonymous referees fortheir insightful comments

BIBLIOGRAPHY

Andreacuteka H Madaraacutesz J X amp Neacutemeti I (2007) Logic of space-time and relativitytheory In Aiello M Pratt-Hartmann I amp Benthem J editors Handbook of SpatialLogics Dordrecht Springer Netherlands pp 607ndash711

Andreacuteka H Madaraacutesz J X Neacutemeti I Stannett M amp Szeacutekely G (2014) Fasterthan light motion does not imply time travel Classical and Quantum Gravity 31(9)095005

Andreacuteka H Madaraacutesz J X Neacutemeti I amp Szeacutekely G (2008) Axiomatizing relativisticdynamics without conservation postulates Studia Logica 89(2) 163ndash186

Andreacuteka H Madaraacutesz J X Neacutemeti I amp Szeacutekely G (2011) A logic road from specialrelativity to general relativity Synthese 186(3) 633ndash649

Borisov Y F (1978) Axiomatic definition of the Galilean and Lorentz groups SiberianMathematical Journal 19(6) 870ndash882

Einstein A (1916) Grundlage der allgemeinen Relativitaumltstheorie Annalen der Physik(ser 4) 49 769ndash822 Available in English translation as Einstein (1996)

Einstein A (1996) The foundation of the general theory of relativity In Klein M JKox A J and Schulman R editors The Collected Papers of Albert Einstein Volume 6The Berlin Years Writings 1914ndash1917 Princeton New Jersey Princeton UniversityPress pp 146ndash200

Friedman M (1983) Foundations of Space-Time Theories Relativistic Physics andPhilosophy of Science Princeton New Jersey Princeton University Press



Galileo (1953) Dialogue Concerning the Two Chief World Systems Berkeley andLos Angeles University of California Press Originally published in Italian 1632Translated by Stillman Drake

Goumlmoumlri M (2015) The Principle of RelativitymdashAn Empiricist Analysis PhD ThesisEoumltvoumls University Budapest

Goumlmoumlri M amp Szaboacute L E (2013a) Formal statement of the special principle of relativitySynthese 192(7) 2053ndash2076

Goumlmoumlri M amp Szaboacute L E (2013b) Operational understanding of the covariance ofclassical electrodynamics Physics Essays 26 361ndash370

Ignatowski W V (1910) Das Relativitaumltsprinzip Archiv der Mathematik und Physik 171ndash24 (Part 1)


Leacutevy-Leblond J-M (1976) One more derivation of the Lorentz transformation AmericanJournal of Physics 44(3) 271ndash277

Madaraacutesz J X (2002) Logic and Relativity (in the Light of Definability Theory) PhDThesis MTA Alfreacuted Reacutenyi Institute of Mathematics

Madaraacutesz J X Stannett M amp Szeacutekely G (2014) Why do the relativistic massesand momenta of faster-than-light particles decrease as their speeds increase SymmetryIntegrability and Geometry Methods and Applications (SIGMA) 10(005) 20

Madaraacutesz J X amp Szeacutekely G (2014) The existence of superluminal particles is consistentwith relativistic dynamics Journal of Applied Logic 12 477ndash500

Marker D (2002) Model Theory An Introduction New York SpringerMendelson E (2015) An Introduction to Mathematical Logic (sixth edition) Boca Raton

London New York CRC PressMisner C Thorne K amp Wheeler J (1973) Gravitation San Francisco W FreemanMolnaacuter A amp Szeacutekely G (2015) Axiomatizing relativistic dynamics using formal thought

experiments Synthese 192(7) 2183ndash2222Muller F A (1992) On the principle of relativity Foundations of Physics Letters 5(6)

591ndash595Pal P B (2003) Nothing but relativity European Journal of Physics 24(3) 315Pelissetto A amp Testa M (2015) Getting the Lorentz transformations without requiring

an invariant speed American Journal of Physics 83(4) 338ndash340Stewart I (2009) Galois Theory (third edition) Boca Raton London New York

Washington DC Chapman amp HallSzaboacute L E (2004) On the meaning of Lorentz covariance Foundations of Physics

Letters 17(5) 479ndash496Szeacutekely G (2013) The existence of superluminal particles is consistent with the

kinematics of Einsteinrsquos special theory of relativity Reports on Mathematical Physics72(2) 133ndash152

ALFRED RENYI INSTITUTE OF MATHEMATICSHUNGARIAN ACADEMY OF SCIENCES

PO BOX 127BUDAPEST 1364 HUNGARY

E-mail madaraszjuditrenyimtahuURL httpwwwrenyihusimmadaraszE-mail szekelygergelyrenyimtahuURL httpwwwrenyihusimturms



DEPARTMENT OF COMPUTER SCIENCETHE UNIVERSITY OF SHEFFIELD

211 PORTOBELLO SHEFFIELD S1 4DP UKE-mail mstannettsheffieldacukURL httpwwwdcsshefacuksimmps



Since all three of the versions we investigate are ldquonaturalrdquo and simply reflect differentapproaches to capturing the original idea there is no point trying to decide which is theldquoauthenticrdquo formalisation The best thing we can do is to investigate how the differentformalisations are related to each other Therefore in this paper we investigate under whichassumptions these formalisations become equivalent

In a different framework but with similar motivations Goumlmoumlri and Szaboacute also intro-duce several formalisations of Einsteinrsquos ideas (Szaboacute 2004 Goumlmoumlri amp Szaboacute 2013abGoumlmoumlri 2015) Intuitively what they refer to as ldquocovariancerdquo corresponds to our principlesof relativity and what they call the principle of relativity is an even stronger assumptionHowever justifying this intuition is beyond the scope of this paper as it would require usto develop a joint framework in which both approaches can faithfully be interpreted

11 Contribution In this paper we present three logical interpretations (SPRMSPR+ and SPRBIOb) of the relativity principle and investigate the extent to which theyare equivalent We find that the three formalisations are logically distinct although theycan be rendered equivalent by the introduction of additional axioms We prove rigorouslythe following relationships

111 Counter-examples and implications requiring no additional axioms (Fig 1)

bull SPR+ rArr SPRM (Theorem 71)bull SPRBIOb rArr SPRM (Theorem 71)bull SPRBIOb rArr SPR+ (Theorem 71)bull SPRM rArr SPR+ (Theorem 72)bull SPRM rArr SPRBIOb (Theorem 73)

112 Adding axioms to make the different formalisations equivalent (Fig 2)

bull SPR+ rArr SPRBIOb assuming AxId AxEv AxIB AxField (Theorem 74)bull SPRBIOb rArr SPRM assuming L = L0 AxEv AxExt (Theorem 77)bull SPR+ rArr SPRM assuming L = L0 AxId AxIB AxField AxEv AxExt (The-

orems 74 77)bull SPRBIOb rArr SPR+ assuming L = L0 AxEv AxExt (Theorems 72 77)

Fig 1 Counter-examples and implications requiring no additional axioms





(Fig 3)































objects)


sorts the formula


)











































and






















)




)













M 2 whenever M 1






























ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY








































(Fig 3)































objects)


sorts the formula


)











































and






















)




)













M 2 whenever M 1






























ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY































































objects)


sorts the formula


)











































and






















)




)













M 2 whenever M 1






























ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY













































objects)


sorts the formula


)











































and






















)




)













M 2 whenever M 1






























ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY











































































and






















)




)













M 2 whenever M 1






























ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY





















































and






















)




)













M 2 whenever M 1






























ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY





















































)




)













M 2 whenever M 1






























ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY













































)




)













M 2 whenever M 1






























ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY








































M 2 whenever M 1






























ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY

















































ϕ(y1 y2 y3 y4 x11 x1

2 x13 x1

4 xn1 xn

2 xn3 xn


11 x1

2 x13 x1

4) (xn1 xn

2 xn3 xn

4 ) isin wlineh(c)]






















































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY


















































































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY







































































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY














































holds ie































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY






















































= wkprimekprime













Proof of injection

























BIBLIOGRAPHY









































Proof of injection

























BIBLIOGRAPHY











































BIBLIOGRAPHY




































































Documents

THREE DIFFERENT FORMALISATIONS OF EINSTEIN’S RELATIVITY