Abstract

In this article, we give a second-order synthetic axiomatization Gal(1, 3) for Galilean spacetime, the background spacetime of Newtonian classical mechanics. The primitive notions of this theory are the 3-place predicate of betweenness Bet, the 2-place predicate of simultaneity ∼, and a 4-place congruence predicate, written ≡^∼, restricted to simultaneity hypersurfaces. We define a standard coordinate structure 𝔾^(1, 3), whose carrier set is ℝ⁴, and which carries relations (on ℝ⁴) corresponding to Bet, ∼, and ≡^∼. This is the standard model of Gal(1, 3). We prove that the symmetry group of 𝔾^(1, 3) is the (extended) Galilean group (an extension of the usual 10-parameter Galilean group with two additional parameters for length and time scalings). We prove that each full model of Gal(1, 3) is isomorphic to 𝔾^(1, 3).

This article provides a synthetic (and second-order) axiom system, which I call $\mathsf{Gal}(1,3)$, which describes Galilean spacetime and does so categorically.¹ Galilean spacetime is a system $\mathbb{P}$ of points on which three physical geometrical primitives are defined, satisfying certain conditions.² Galilean spacetime can be thought of as the background geometry of the system of spacetime events for Newtonian classical mechanics:

I shall call the carrier set of Galilean spacetime $\mathbb{P}$: this is the domain of “spacetime points” or “events.” Going ahead of ourselves a bit, there are three distinguished physical relations on $\mathbb{P}$. A three-place betweenness relation $B$, which gives the whole system an affine “straight-line” structure;³ a binary simultaneity relation $\sim$, which induces a partition of $\mathbb{P}$ into a system of non-intersecting simultaneity hypersurfaces, $\Sigma_{0},\Sigma_{1},\dots$, arranged as a “foliation”; and a special four-place congruence relation: this is the four-place sim-congruence relation, $\equiv^{\sim}$, which induces three-dimensional Euclidean geometry on each hypersurface.⁴

An especially important subset of straight lines are “time axes”: a time axis is a straight line in the affine geometry that does not lie within a simultaneity hypersurface. Physically, a time axis is the trajectory of a material point acted on by no forces—this is Newton’s First Law or the Law of Inertia.⁵

We can bundle the carrier set of Galilean spacetime and the aforementioned three distinguished physical relations on Galilean spacetime together: $(\mathbb{P},B,\sim,\equiv^{\sim})$. Our aim in this paper is to give a synthetic axiomatization of this structure $(\mathbb{P},B,\sim,\equiv^{\sim})$.⁶ This means that, in contrast with analytic geometry, the axioms do not quantify over the reals, introduce a metric function (like a Riemannian metric $g_{ab}$), or talk about coordinate systems. Instead, the axioms use a number of basic physical predicates on spacetime. And then the existence of special mappings $\Phi:\mathbb{P}\to\mathbb{R}^{4}$—that is, coordinate systems—becomes a theorem, not an assumption.

Hartry Field (1980) has carefully studied this approach in order to try and vindicate nominalism: this is the claim that there are no mathematical objects at all, and insofar as numbers, functions, sets, vector spaces, Lie groups, and so on are used in physics and science more generally, they can be dispensed with. It is the claim that physical theories can, in principle, be replaced with theories that are “nominalistic” and that the normal use of mathematics is “useful but false.” It is to Field’s enormous credit to have pinned down the two essential uses. These are:

Expressiveness. We can express physical laws by, e.g., “$\nabla\cdot\mathbf{B}=0$” and so on. So, $\mathbf{B}$ is a mixed function that maps each point to some numbers. As Feynman put it, “From a mathematical view, there is an electric field vector and a magnetic field vector at every point in space; that is, there are six numbers associated with every point” (Feynman, Leighton and Sands 2005, chaps. 20, sec.3).

Proof-Theoretic. Mathematically reasoning is generally conservative over non-mathematical premises, but using mathematics, we can get “quicker proofs” of a non-mathematical conclusion C from a non-mathematical premise P.

As regards the second, in mathematical logic, this is called “speed-up,” and it was discovered by Kurt Gödel (1935) as a spin-off from his incompleteness results. Perhaps the most remarkable example of this phenomenon was given in Boolos (1987), a first-order valid inference with a short mathematical proof (it uses second-order comprehension), but whose shortest purely logical derivation, using the rules for the connectives and quantifiers, has vastly more symbols than the number of baryons in the observable universe.⁷

The best survey, and overall evaluation, of a large variety of nominalist approaches for both mathematics and science is Burgess and Rosen (1997).⁸ I’m not recommending this as an approach to studying the geometrical assumptions of physical theories, as my own view here is the usual mathematical realist view (“useful because true”). Indeed, Riemannian geometry is here to stay! Riemannian geometry provides incredible flexibility by assuming the existence of a metric tensor $g_{ab}$ on spacetime.⁹ However, for the two special cases of Galilean spacetime and Minkowski spacetime, the synthetic approach helps provide a nice example of how the physics (i.e., the basic physical relations: betweenness, congruence, and so on) and mathematics (i.e., real numbers, coordinate systems, vector spaces, and so on) get “entangled.”

The basic machinery for the introduction of coordinates is the Representation Theorem. Given a synthetic structure satisfying a series of conditions, one proves the existence of an isomorphism to a standard coordinate structure:¹⁰

\[\Phi:\text{synthetic structure}\to\text{coordinate structure}. \qquad{(1)}\]

That is, the isomorphism $\Phi$ takes each point $p$ in the synthetic structure to its coordinates $\Phi^i(p)$ (usually in $\mathbb{R}^{n}$) in such a way that a distinguished synthetic relation $R$ holds for $p,q,\dots$ iff a separately defined coordinate relation $R'$ holds for $\Phi(p),\Phi(q),\dots$ (see, for example, (5) below). Because the synthetic and coordinate structures are isomorphic, the latter is a kind of map or representation of the former: they share the same abstract structure.¹¹

However, historically, the analysis of Galilean spacetime did not proceed like this. Modern analysis of Galilean spacetime (sometimes called “neo-Newtonian” spacetime or just “Newtonian spacetime”) was developed using the differential geometry methods developed to study General Relativity: what are now called “relativistic spacetimes.” This began in the 60s and 70s, with work by Trautman, Penrose, Stein, Ehlers, Earman, and others (based on earlier work, such as Cartan’s).¹² In Malament (2012, chap. 4), David Malament provides details of the differential geometry formulation of this topic. Galilean (or Newtonian) spacetime is defined as a structure of the form

\[\mathcal{A}=(M,\nabla,h^{ab},t_{ab}), \qquad{(2)}\]

where $M$ is a manifold diffeomorphic to $\mathbb{R}^{4}$, $\nabla$ is a flat (torsion-free) affine connection on $M$, and $h^{ab},t_{ab}$ are tensor fields on $M$ satisfying compatibility conditions, from which one can construct temporal and spatial metrics and simultaneity surfaces.¹³

The approach we develop here is entirely synthetic. The underlying geometric relations are betweenness (written $\texttt{Bet}(p,q,r)$), simultaneity (written $p\sim q$), and sim-congruence (written $pq\equiv^{\sim}rs$): these are relations on points. Inertial coordinate systems are then proved to exist by a Representation Theorem. An inertial coordinate system $\Phi$ is nothing more than an isomorphism from the synthetic geometrical structure $(\mathbb{P},B,\sim,\equiv^{\sim})$ of Galilean spacetime (with carrier set $\mathbb{P}$) to a suitable “coordinate structure” built on the carrier set $\mathbb{R}^{4}$. Below, we shall call this standard coordinate structure $\mathbb{G}^{(1,3)}$ (Definition 4). So, we shall obtain, by analogy with (1),

\[\Phi:\overbrace{(\mathbb{P},B,\sim,\equiv^{\sim})}^{\text{synthetic structure}}\to\overbrace{\ \mathbb{G}^{(1,3)}.}^{\text{coordinate structure}}\qquad{(3)}\]

Euclidean geometry, of course, was also first set out synthetically in Euclid’s Elements. However, Euclid’s Elements does not quite meet modern adequate standards of formal rigor. In particular, Moritz Pasch (1882) noted that certain betweenness properties of space were merely implicit in Euclid’s treatment. Influenced by Pasch and others, the synthetic axiomatization for Euclidean geometry was first made rigorous in Hilbert (1899), which was modified, extended, or simplified in a number of ways, one of which is Veblen (1904) (which extracted the purely betweenness part of Hilbert’s system: sometimes called the “axioms of order”).

Synthetic axiomatization for Minkowski spacetime geometry appeared soon after the classic work of Albert Einstein and Hermann Minkowski (i.e., Einstein 1905; Minkowski 1909) in Alfred Robb’s (1911) book. This led to a series of later synthetic developments, including Robb (1936), Ax (1978), Mundy (1986), Goldblatt (1987), Schutz (1997), and, most recently, Cocco and Babic (2021). As is now known, Minkowski spacetime can be axiomatized using a single binary relation, usually called $\lambda$, with $p\lambda q$ meaning “points $p$ and $q$ can be connected by a light signal”—the light-signal relation.¹⁴ As the reader probably knows, this induces a “light cone structure” on the carrier set of points. So, Minkowski spacetime can be defined as a structure $(\mathbb{P},\lambda)$ satisfying certain axioms, and one may prove that there is an isomorphism $\Phi:(\mathbb{P},\lambda)\to(\mathbb{R}^{4},\lambda_{\mathbb{R}^{4}})$.¹⁵ Such an isomorphism is called a “Lorentz coordinate system.” Then the automorphism group $\text{Aut}((\mathbb{R}^{4},\lambda_{\mathbb{R}^{4}}))$ of $(\mathbb{R}^{4},\lambda_{\mathbb{R}^{4}})$ is the Poincaré group.¹⁶

Galilean spacetime, however, is the basic spacetime of classical Newtonian (pre-relativistic) physics. In retrospect, it is a kind of “low energy limit” of Minkowski spacetime (when we let the speed of light approach infinity and all the light cones get “squashed” into simultaneity surfaces). But, unlike the case with Minkowski spacetime, the synthetic approach did not appear for a long time. As far as I know, the first brief sketch of a synthetic axiom system for Galilean spacetime appeared in Hartry Field’s Science Without Numbers (1980, chap. 6), some 80 years after Hilbert’s classic monograph, The Foundations of Geometry (1899), and close on three hundred years after Newton’s Principia (1687). Shortly after, John Burgess added further work on this in Burgess (1984) and then again in Burgess and Rosen (1997). Our work here is a descendant of and stimulated by theirs.¹⁷

The axiom system $\mathsf{Gal}(1,3)$ we shall arrive at can be written as follows (see table 1 in section 3):

$\mathsf{Gal}1$	$\mathsf{BG}(4)$.
$\mathsf{Gal}2$	$\mathsf{EG}(3)^{\sim}$.
$\mathsf{Gal}3$	$\sim$ is an equivalence relation.
$\mathsf{Gal}4$	$\equiv^{\sim} \ \subseteq \ [\sim]^{4}$.
$\mathsf{Gal}5$	$\equiv^{\sim}$ is translation-invariant.

Here, $\mathsf{BG}(4)$ is a group of nine axioms, the subsystem of order axioms for betweenness (see appendix 5.1 below). And $\mathsf{EG}(3)^{\sim}$ is a group of eleven axioms, a relativized subsystem of axioms for “sim-congruence” and betweenness, obtained from Tarski’s formulation of Euclidean geometry for three dimensions (see appendix 5.1). The three further axioms, $\mathsf{Gal}3$, $\mathsf{Gal}4$, and $\mathsf{Gal}5$, “tie together” these subsystems.¹⁸

To summarize, then, how the rest of this paper goes, we shall use the two separate Representation Theorems for $\mathsf{BG}(4)$ and $\mathsf{EG}(3)$. The first of these (theorem 62 in appendix 5.2 below) asserts the existence of a “global” bijective coordinate system:

\[\Phi:\mathbb{P}\to\mathbb{R}^{4},\qquad{(4)}\]

on any (full) model $(\mathbb{P},B)$ of $\mathsf{BG}(4)$, matching any given “4-frame” $O,X,Y,Z,I$ and satisfying the betweenness representation condition, for any points $p,q,r\in\mathbb{P}$:¹⁹

\[\begin{aligned} B(p,q,r) & \leftrightarrow B_{\mathbb{R}^{4}}(\Phi(p),\Phi(q),\Phi(r)), \end{aligned}\qquad{(5)}\]

where $B_{\mathbb{R}^{4}}$ is the standard betweenness relation on $\mathbb{R}^{4}$. The second Representation Theorem (theorem 63 in appendix 5.2) asserts the existence of a global coordinate system $\psi$ on any (full) model $(\mathbb{P},B,\equiv)$ of three-dimensional Euclidean geometry $\mathsf{EG}(3)$, matching a given “Euclidean 3-frame” $O,X,Y,Z$ and satisfying the representation condition for congruence:

\[\begin{aligned} pq\equiv rs & \leftrightarrow\psi(p)\psi(q)\equiv_{\mathbb{R}^{3}}\psi(r)\psi(s), \end{aligned}\qquad{(6)}\]

where $\equiv_{\mathbb{R}^{3}}$ is the standard congruence relation on $\mathbb{R}^{3}$. In our system, the axioms $\mathsf{EG}(3)$ are relativized to simultaneity hypersurfaces, yielding $\mathsf{EG}(3)^{\sim}$. The relativization implements the requirement that each simultaneity hypersurface is a three-dimensional Euclidean space.

We can then combine these two Representation Theorems, applied to any full model $M\models_{2}\mathsf{Gal}(1,3)$, to obtain the Representation Theorem for $\mathsf{Gal}(1,3)$, which is our main theorem (theorem 55 in section 5). That is, assuming $(\mathbb{P},B,\sim,\equiv^{\sim})$ is a (full) model of $\mathsf{Gal}(1,3)$, the existence of an isomorphism as stated in (3) above:

\[\Phi:\overbrace{(\mathbb{P},B,\sim,\equiv^{\sim})}^{\text{synthetic structure}}\to\overbrace{\ \mathbb{G}^{(1,3)}.\ }^{\text{coordinate structure}}\qquad{(7)}\]

The crux of the proof of the main theorem are the Chronology Lemma (lemma 52) and the Congruence Lemma (lemma 54).

1 Definitions

Definition 1. The standard Euclidean inner product $\langle ., .\rangle_{n}$ and norm $\|.\|_{n}$ on $\mathbb{R}^{n}$ are defined as follows:²⁰ For $\mathbf{x},\mathbf{y}\in\mathbb{R}^{n}$, $\langle\mathbf{x},\mathbf{y}\rangle_{n}:=\sum_{i=1}^{n}x^{i}y^{i}$, and $\|\mathbf{x}\|_{n}:=\sqrt{\langle\mathbf{x},\mathbf{x}\rangle_{n}}$. The standard Euclidean metrics $\Delta_{n}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ are defined as follows:

\[\begin{aligned} \Delta_{n}(\mathbf{x},\mathbf{y}) & :=\|\mathbf{x}-\mathbf{y}\|_{n}. \end{aligned}\qquad{(8)}\]

The standard Euclidean metric space with carrier set $\mathbb{R}^{n}$ is:

\[\begin{aligned} \mathbb{EG}_{\text{metric}}^{n} & :=(\mathbb{R}^{n},\Delta_{n}). \end{aligned}\qquad{(9)}\]

Definition 2. The following relations are the standard betweenness relation $B_{\mathbb{R}^{n}}$, standard simultaneity relation $\sim_{\mathbb{R}^{n}}$, standard congruence relation $\equiv_{\mathbb{R}^{n}}$, and standard sim-congruence relation $\equiv^{\sim}_{\mathbb{R}^{n}}$ on $\mathbb{R}^{n}$. For $\mathbf{x},\mathbf{y},\mathbf{z},\mathbf{u}\in\mathbb{R}^{n}$:

\[\begin{aligned} B_{\mathbb{R}^{n}}(\mathbf{x},\mathbf{y},\mathbf{z}) & :=(\exists\lambda\in[0,1])(\mathbf{y}-\mathbf{x}=\lambda(\mathbf{z}-\mathbf{x})); & (a)\\ \mathbf{x}\sim_{\mathbb{R}^{n}}\mathbf{y} & :=x^{n}=y^{n}; & (b)\\ \mathbf{x}\mathbf{y}\equiv_{\mathbb{R}^{n}}\mathbf{z}\mathbf{u} & :=\Delta_{n}(\mathbf{x},\mathbf{y})=\Delta_{n}(\mathbf{z},\mathbf{u}); & (c)\\ \mathbf{x}\mathbf{y}\equiv^{\sim}_{\mathbb{R}^{n}}\mathbf{z}\mathbf{u} & :=\Delta_{n}(\mathbf{x},\mathbf{y})=\Delta_{n}(\mathbf{z},\mathbf{u})\ \&\ \mathbf{x}\sim_{\mathbb{R}^{n}}\mathbf{y}\ \&\ \mathbf{x}\sim_{\mathbb{R}^{n}}\mathbf{z}\ \&\ \mathbf{x}\sim_{\mathbb{R}^{n}}\mathbf{u}. & (d)\\ \end{aligned}\qquad{(10)}\]

For the one-dimensional case, we have two alternative but equivalent definitions. First, $B_{\mathbb{R}}(x,y,z):=(x\leq y\leq z)$; second, $B_{\mathbb{R}}(x,y,z):=|x-y|+|y-z|=|x-z|$.²¹

Definition 3. It will be useful below to define the following special five points in $\mathbb{R}^{4}$:

\[\begin{aligned} \mathbf{O}:=\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0\end{pmatrix},\quad \mathbf{X}:=\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0\end{pmatrix},\quad \mathbf{Y}:=\begin{pmatrix} 0 \\ 1 \\ 0 \\ 0\end{pmatrix},\quad \mathbf{Z}:=\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0\end{pmatrix},\quad \mathbf{I}:=\begin{pmatrix} 0 \\ 0 \\ 0 \\ 1\end{pmatrix}. \end{aligned}\qquad{(11)}\]

In other words, the origin and the “unit points” on the four axes. I call the ordered tuple $\mathbf{O},\mathbf{X},\mathbf{Y},\mathbf{Z},\mathbf{I}$ the standard (4-)frame in $\mathbb{R}^{4}$.

Definition 4. The standard coordinate structures are:²²

$\mathbb{BG}^{n}$	Betweenness geometry in $n$ dimensions over $\mathbb{R}$	$:=(\mathbb{R}^{n},B_{\mathbb{R}^{n}}).$
$\mathbb{EG}^{n}$	Euclidean space in $n$ dimensions over $\mathbb{R}$	$:=(\mathbb{R}^{n},B_{\mathbb{R}^{n}},\equiv_{\mathbb{R}^{n}}).$
$\mathbb{G}^{(1,n)}$	Galilean spacetime in $n+1$ dimensions over $\mathbb{R}$	$:=(\mathbb{R}^{n+1},B_{\mathbb{R}^{n+1}},\sim_{\mathbb{R}^{n+1}},\equiv^{\sim}_{\mathbb{R}^{n+1}}).$

Our central interest is $\mathbb{G}^{(1,3)}$, the standard coordinate structure for four-dimensional Galilean spacetime. The carrier set of $\mathbb{G}^{(1,3)}$ is $\mathbb{R}^{4}$. Its distinguished relations are betweenness (10, a), simultaneity (10, b), and sim-congruence (10, d) on $\mathbb{R}^{4}$. Note that $\mathbb{G}^{(1,3)}$ does not carry a metric or distance function.

2 Derivation of (Extended) Galilean Transformations

What is the symmetry group of the standard coordinate structure $\mathbb{G}^{(1,3)}$ for Galilean spacetime? We will see that its symmetry group is a certain Lie group $\mathcal{G}^{e}(1,3)$, a 12-dimensional Lie group that extends the usual Galilean group $\mathcal{G}(1,3)$ by two additional parameters, which determine coordinate scalings.

Definition 5. $A$ is an element of the extended Galilean matrix group $\mathsf{Mat}^{e}_{\mathsf{Gal}}(4)$ if and only if $A$ is a $4 \times 4$ matrix with real entries and has the (block matrix) form

\[\begin{aligned} A=\begin{pmatrix}\alpha_{1}R & \vec{v} \\ 0 & \alpha_{2}\end{pmatrix} \end{aligned},\qquad{(12)}\]

where

\[\begin{aligned} R=\begin{pmatrix}R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33}\end{pmatrix} \end{aligned}\qquad{(13)}\]

is in $O(3)$, $\vec{v}=(v_{1},v_{2},v_{3})\in\mathbb{R}^{3}$, and $\alpha_{1},\alpha_{2}\in\mathbb{R}-\{0\}$. The $O(3)$ matrix $R$ is called the rotation of $A$, the 3-vector $\vec{v}$ is called the (relative) velocity of $A$, the constant $\alpha_{1}$ is called the spatial scaling factor of $A$, and the constant $\alpha_{2}$ is the temporal scaling factor of $A$.

Lemma 6. $\mathsf{Mat}^{e}_{\mathsf{Gal}}(4)$ is a subgroup of $GL(4)$.

Proof. This is a routine verification. The main part is to check that $\mathsf{Mat}^{e}_{\mathsf{Gal}}(4)$ is closed under matrix multiplication and each element in $\mathsf{Mat}^{e}_{\mathsf{Gal}}(4)$ has an inverse in $\mathsf{Mat}^{e}_{\mathsf{Gal}}(4)$.∎

Definition 7. Let $h:\mathbb{R}^{4}\to\mathbb{R}^{4}$. We say that $h$ is an extended Galilean transformation just if there exists an extended Galilean matrix $A$ and a displacement $\mathbf{d}\in\mathbb{R}^{4}$ such that, for all $\mathbf{x}\in\mathbb{R}^{4}$,

\[h(\mathbf{x})=A\mathbf{x}+\mathbf{d}.\qquad{(14)}\]

Lemma 8. The set of extended Galilean transformations forms a group.

Proof. This is a detailed verification of the group properties, analogous to the above.∎

Definition 9. $\mathcal{G}^{e}(1,3):=$ the group of extended Galilean transformations.

Theorem 10 (Automorphisms of $\mathbb{G}^{(1,3)}$). $\text{Aut}(\mathbb{G}^{(1,3)})=\mathcal{G}^{e}(1,3)$.

Proof. I give a sketch of the proof. To show $\mathcal{G}^{e}(1,3)\subseteq\text{Aut}(\mathbb{G}^{(1,3)})$, we verify that each extended Galilean transformation is a symmetry of $\mathbb{G}^{(1,3)}$. Since $\mathbb{BG}^{4}$ is a reduct of $\mathbb{G}^{(1,3)}$, and each extended Galilean transformation is affine, it follows that betweenness is invariant. The special form of extended Galilean matrices then ensures that simultaneity and sim-congruence are invariant.

To show that $\text{Aut}(\mathbb{G}^{(1,3)})\subseteq\mathcal{G}^{e}(1,3)$ is more involved. Since $\mathbb{BG}^{4}$ is a reduct of $\mathbb{G}^{(1,3)}$, it follows that any symmetry $h$ of $\mathbb{G}^{(1,3)}$ must be affine, and so there exists a $GL(4)$ matrix $A$ and displacement $\mathbf{d}\in\mathbb{R}^{4}$ such that, for any $\mathbf{x}\in\mathbb{R}^{4}$,

\[h(\mathbf{x})=A\mathbf{x}+\mathbf{d}.\qquad{(15)}\]

To determine the sixteen components $A_{ij}$ of $A$, one must then examine the conditions that simultaneity and sim-congruence be invariant. By examining certain choices of points, the invariance of simultaneity enforces that $A$ must have the form

\[A=\begin{pmatrix}C & \vec{v} \\ 0 & \alpha_{2}\end{pmatrix},\qquad{(16)}\]

where $C$ is a $3\times 3$ matrix, and $\alpha_{2}$ is a non-zero constant. The invariance of sim-congruence enforces that the upper $3\times 3$ block $C$ must be a multiple $\alpha_{1}R$ of an $O(3)$ matrix $R$ by a non-zero real factor $\alpha_{1}$:

\[A=\begin{pmatrix}\alpha_{1} R & \vec{v} \\ 0 & \alpha_{2}\end{pmatrix}.\qquad{(17)}\]

But this is an extended Galilean matrix. Consequently, $\text{Aut}(\mathbb{G}^{(1,3)})\subseteq\mathcal{G}^{e}(1,3)$.

Together, these results imply that $\text{Aut}(\mathbb{G}^{(1,3)})=\mathcal{G}^{e}(1,3)$.∎

The constants $\alpha_{1},\alpha_{2}$ in any extended Galilean matrix $A$ determine scalings of the spatial and temporal coordinates, respectively. So, given some $A$ in the extended Galilean matrix group and any $(\vec{x},t)\in\mathbb{R}^{4}$,

\[A(\vec{x},t)=(\alpha_{1}R\vec{x}+\vec{v}t,\alpha_{2}t).\qquad{(18)}\]

Let’s set the relative rotation $R$ to be $\mathbb{I}$ and set the relative velocity $\vec{v}$ to be zero:

\[A(\vec{x},t)=(\alpha_{1}\vec{x},\alpha_{2}t).\qquad{(19)}\]

Thus, the spatial coordinates are scaled by $\alpha_{1}$, and the temporal coordinate is scaled by $\alpha_{2}$. Instead, let us set these scalings $\alpha_{1},\alpha_{2}$ at $1$ and consider the image $(\vec{x}',t')$ of the point with coordinates $(\vec{x},t)$ under an extended Galilean transformation:

\[\vec{x}'=R\vec{x}+\vec{v}t+\vec{d},\qquad{(20)}\] \[t'=t+d_{t}.\qquad{(21)}\]

These are the usual Galilean transformations as given in physics textbooks, in usually simplified form (e.g., Sears, Zemansky and Young 1979, 252; Longair 1984, 87; or Rindler 1977, 3). The conventional Galilean group $\mathcal{G}(1,3)$ is normally understood to be this 10-parameter Lie group: the ten parameters are these: four parameters for the spatial and temporal translations, $\mathbf{d}$; three parameters (i.e., determined by the three Euler angles) for the rotation matrix $R$; three parameters for the velocity $\vec{v}$.

As we have defined it, the extended Galilean group $\mathcal{G}^{e}(1,3)$ is a 12-parameter Lie group: the two additional parameters, $\alpha_{1},\alpha_{2}$, permit coordinate scalings. These two extra degrees of freedom are a consequence of our synthetic treatment, and this is completely analogous to Euclidean betweenness and congruence being invariant under coordinate scaling. Indeed, $\alpha_{1}$ and $\alpha_{2}$ are gauge parameters in the oldest sense of the word.

3 Axiomatization of Galilean Spacetime: $\mathsf{Gal}(1,3)$

To begin, we state the informal physical meanings of our three primitive symbols:²³

Betweenness Predicate: $\texttt{Bet}$. $\texttt{Bet}(p,q,r)$ means that $q$ lies on a straight line inclusively between $p$ and $r$ (allowing the cases $q=p$ and $q=r$).

Simultaneity Predicate: $\sim$. $p \sim q$ means that the points $p,q$ are simultaneous.

Sim-Congruence Predicate: $\equiv^{\sim}$. $pq\equiv^{\sim}rs$ means the points $p,q,r,s$ are simultaneous, and the length of the segment $pq$ is equal to the length of the segment $rs$.

We are now ready to state the (synthetic) axioms for Galilean spacetime.

Definition 11. The theory $\mathsf{Gal}(1,3)$ is a two-sorted theory with sorts $\{\texttt{point},\texttt{point}\texttt{set}\}$ and variables $\textsf{Var}_{\texttt{point}}=\{p_{1},p_{2},\dots\}$ and $\textsf{Var}_{\texttt{point}\texttt{set}}=\{\mathsf{X}_{1},\mathsf{X}_{2},\dots\}$. The signatures $\sigma_{\mathsf{Gal}}$ and $\sigma_{\mathsf{Gal},\in}$ are given by $\sigma_{\mathsf{Gal}}=\{\texttt{Bet},\sim,\equiv^{\sim}\}$ and $\sigma_{\mathsf{Gal},\in}=\{\texttt{Bet},\sim,\equiv^{\sim},\in\}$. By $L(\sigma_{\mathsf{Gal}})$, I shall mean the first-order language with restricted signature $\sigma_{\mathsf{Gal}}$ over the single sort $\texttt{point}$. Its atomic formulas are of the four forms: $p_{1}=p_{2}$, $\texttt{Bet}(p_{1},p_{2},p_{3})$, $p_{1}\sim p_{2}$, and $p_{1}p_{2}\equiv^{\sim}p_{3}p_{4}$, where “$p_{i}$” are point variables, and the remaining formulas are built up using the connectives $\neg,\wedge,\vee,\to,\leftrightarrow$, and quantifiers $\forall$ and $\exists$, as per the usual recursive definition of “formula of $L(\sigma)$.”²⁴ By $L(\sigma_{\mathsf{Gal},\in})$, I mean the “monadic second-order” language, with signature $\sigma_{\mathsf{Gal},\in}$. Its atomic sentences include those above along with formulas: $p_{i}\in\mathsf{X}_{j}$ and $\mathsf{X}_{i}=\mathsf{X}_{j}$. (A parser for this language counts the strings $p_{i}=\mathsf{X}_{j}$, $\mathsf{X}_{j}=p_{i}$, and $\mathsf{X}_{i}\in p_{j}$ and $p_{i}\in p_{j}$ as ill-formed.) The remaining formulas are built up using the connectives $\neg,\wedge,\vee,\to,\leftrightarrow$, and quantifiers $\forall$ and $\exists$, including the new quantifications $\forall\mathsf{X}_{i}\,\varphi$ and $\exists\mathsf{X}_{i}\,\varphi$.

In discussing a full model $M$ of, say, $\mathsf{BG}(4)$, I shall generally write “$M\models_{2}\mathsf{BG}(4)$” to make it clear that $M$ is a full model of $\mathsf{BG}(4)$. In other words, if $M=(\mathbb{P},\dots)$, then $M\models_{2}\forall\mathsf{X}_{i}\,\varphi(\mathsf{X}_{i})$ if and only if, for every subset $U\subseteq\mathbb{P}$, $\varphi[U]$ is true in $M$.

Definition 12. $(p,q,r,s)\in[\sim]^{4}$ iff $p\sim q,p\sim r,p\sim s$.

Definition 13. The (non-logical) axioms of $\mathsf{Gal}(1,3)$ are as follows:

Table 1: The axiom system $\mathsf{Gal}(1,3)$.
$\mathsf{Gal}1$	$\mathsf{BG}(4)$.
$\mathsf{Gal}2$	$\mathsf{EG}(3)^{\sim}$.
$\mathsf{Gal}3$	$\sim$ is an equivalence relation.
$\mathsf{Gal}4$	$\equiv^{\sim} \ \subseteq \ [\sim]^{4}$.
$\mathsf{Gal}5$	$\equiv^{\sim}$ is translation-invariant.

$\mathsf{BG}(4)$ is really an axiom group of nine axioms for $\texttt{Bet}$.²⁵ These are given in definition 56 in appendix 5.1. But, to simplify the description here, one may take their conjunction.²⁶ $\mathsf{EG}(3)$ is also an axiom group, this time of eleven axioms. These are given in definition 57 in appendix 5.1. The axiom $\mathsf{EG}(3)^{\sim}$ listed above requires further explanation.²⁷

This construction is sketched, very briefly, in Field (1980, 54, n.33). First, one replaces $\equiv$ by $\equiv^{\sim}$ in each $\mathsf{EG}(3)$ axiom. Next, one relativizes each axiom to the formula $p\sim z$ (treating $z$ as a parameter) so that the resulting axiom says that it holds for all points simultaneous with $z$.²⁸ Next, one prefixes the result with $\forall z$ and then takes the conjunction of the axioms. For example, under relativization, the $\equiv$-Transitivity axiom (E3) and the Pasch axiom (E6) become:

$\equiv^{\sim}$-Transitivity	$\forall z[(\forall p,q,r,s,t,u\sim z)\,(pq\equiv^{\sim}rs\wedge pq\equiv^{\sim}tu\to rs\equiv^{\sim}tu)]$.
Pasch	$\forall z[(\forall p,q,r,s,u\sim z)\,(\texttt{Bet}(p,q,r)\wedge\texttt{Bet}(s,u,q) \to(\exists x\sim z)\,(\texttt{Bet}(r,x,s)\wedge\texttt{Bet}(p,u,x)))]$.

In addition to the given non-logical axioms, we also have the customary axioms for second-order logic (table 2):

Table 2: Axioms for second-order logic.
Comprehension	$\exists\mathsf{X}_{1}\,\forall p\,(p\in\mathsf{X}_{1}\leftrightarrow\varphi)$ (variable $\mathsf{X}_{1}$ not free in $\varphi$).
Extensionality	$\forall\mathsf{X}_{1}\,\forall\mathsf{X}_{2}\,(\forall p\, (p\in\mathsf{X}_{1}\leftrightarrow p\in\mathsf{X}_{2}) \to\mathsf{X}_{1}=\mathsf{X}_{2})$.

I shall, in effect, however, assume an ambient set theory.²⁹ The reason is that I am not concerned with narrow proof-theoretic matters concerning the whole theory (for example, completeness) but rather with establishing some facts about the full models of the theory $\mathsf{Gal}(1,3)$. Since we consider just full models, Comprehension and Extensionality are satisfied more or less by fiat.³⁰ This is completely analogous to our approach in giving the usual proof, essentially that of Dedekind (1888), of the categoricity of second-order arithmetic $\mathsf{PA}_2$, although, as a matter of fact, the categoricity of $\mathsf{PA}_2$ can be “internalized” as a proof inside $\mathsf{PA}_2$ itself (see Simpson and Yokoyama 2013).

The three Galilean axioms $\mathsf{Gal}3$, $\mathsf{Gal}4$, and $\mathsf{Gal}5$ are the glue that holds together the betweenness axioms $\mathsf{BG}(4)$ and the Euclidean axioms $\mathsf{EG}(3)^{\sim}$. The content of $\mathsf{Gal}3$ and $\mathsf{Gal}4$ seems evident. The final axiom $\mathsf{Gal}5$ is the sole axiom that needs some further explanation.³¹ This axiom expresses the translation invariance of the $\equiv^{\sim}$ relation and may be expressed using vector notation as follows:

\[pq\equiv^{\sim}rs\to(p+\mathbf{v})(q+\mathbf{v})\equiv^{\sim}(r+\mathbf{v})(s+\mathbf{v}).\qquad{(22)}\]

In other words, if the (simultaneous) segments $pq$ and $rs$ have the same length, then the (simultaneous) segments $(p+\mathbf{v})(q+\mathbf{v})$ and $(r+\mathbf{v})(s+\mathbf{v})$ have the same length for any vector $\mathbf{v}$.³²

An equivalent axiom can be expressed solely using the primitives $\texttt{Bet}$, $\sim$, and $\equiv^{\sim}$ and quantifying over points. Roughly, the axiom $\mathsf{Gal}5$ is equivalent to the following rather long-winded claim:

If $p,q,r,s$, and $p',q',r',s'$ are points such that the vectors $\mathbf{v}_{p,p'}$, $\mathbf{v}_{q,q'}$, $\mathbf{v}_{r,r'}$, $\mathbf{v}_{s,s'}$ are all equal and $pq\equiv^{\sim}rs$, then $p'q'\equiv^{\sim}r's'$.

Note that the equality clause “$\mathbf{v}_{p,p'}=\mathbf{v}_{q,q'}$” means “$p,q,p',q'$ is a parallelogram,” and the 4-place predicate “$p_{1},p_{2},p_{3},p_{4}$ is a parallelogram” can be defined using $\texttt{Bet}$ (see definition 15).

The second-order theories $\mathsf{BG}(4)$ and $\mathsf{EG}(3)$, with their point set variables, contain the second-order Continuity Axiom (Tarski 1959, 18):

\[\begin{gathered} [\exists r\,(\forall p\in\mathsf{X}_{1})\,(\forall q\in\mathsf{X}_{2}) \texttt{Bet}(r,p,q)]\\ \to[\exists s\,(\forall p\in\mathsf{X}_{1})\,(\forall q\in\mathsf{X}_{2})\,\texttt{Bet}(p,s,q)]. \end{gathered}\qquad{(23)}\]

This geometrical continuity axiom, it may be noted, is closely analogous to the “Dedekind Cut Axiom,” which may be used as an axiom in the formalization of the second-order theory $\mathsf{ALG}$ of real numbers:³³

\[\begin{gathered} (\forall\mathsf{X}_{1}\subseteq\mathbb{R})\,(\forall\mathsf{X}_{2}\subseteq\mathbb{R})\,(\mathsf{X}_{1}\neq\varnothing\wedge\mathsf{X}_{2}\neq\varnothing\wedge\overbrace{(\forall x\in\mathsf{X}_{1})\,(\forall y\in\mathsf{X}_{2})\,(x\leq y)}^{\mathsf{X}_{1}\text{ ``precedes" }\mathsf{X}_{2}}\\ \to\exists s\,\overbrace{(\forall x\in\mathsf{X}_{1})\,(\forall y\in\mathsf{X}_{2})\,(x\leq s\wedge s\leq y))}^{\text{the point $s$ ``cuts" $\mathsf{X}_{1}$ and $\mathsf{X}_{2}$}}. \end{gathered}\qquad{(24)}\]

The second-order theories $\mathsf{BG}(4)$ and $\mathsf{EG}(3)$ are, foundationally speaking, strong, and both interpret $\mathsf{ALG}$. They have first-order versions—their “little brothers,” so to speak, which I shall call $\mathsf{BG}_{0}(4)$ and $\mathsf{EG}_{0}(3)$—obtained by replacing the single Continuity Axiom by infinitely many instances of the Continuity axiom scheme: in these instances, there are only point variables.

The little brothers, $\mathsf{BG}_{0}(4)$ and $\mathsf{EG}_{0}(3)$, are meta-mathematically somewhat different from their big brothers. In particular, they are, in fact, complete (and, since they are recursively axiomatized, decidable), as established by a celebrated theorem of Alfred Tarski (1951). But the big brothers are incomplete because they interpret Peano arithmetic ($\mathsf{PA}$), and then Gödel’s incompleteness results apply. This observation leads to an important difficulty faced by Field’s nominalism:

Remark 14. The second-order nature of $\mathsf{BG}(4)$—i.e., its point variables range over points, and its set variables range over sets of points—is what lies at the root of the technical problem for Hartry Field’s nominalist program (1980) highlighted, first informally by John Burgess, Saul Kripke, and Yiannis Moschovakis, and then, in detail, by Stewart Shapiro in (1983), and also mentioned in Burgess (1984, last section). The required representation theorem indeed holds for $\mathsf{BG}(4)$ with respect to full models (and from this, the other representation theorems can be built up, just as we do below). This is theorem 62 below. But, unfortunately, adding additional set theory axioms to $\mathsf{BG}(4)$ is non-conservative. This is because $\mathsf{BG}(4)$ interprets Peano arithmetic. And then, by Gödel’s incompleteness results (Gödel 1931; Raatikainen 2015), there is a consistency sentence $\textsf{Con}(\mathsf{BG}(4))$ in the language of $\mathsf{BG}(4)$ itself such that $\mathsf{BG}(4)$ does not prove $\textsf{Con}(\mathsf{BG}(4))$. $\textsf{Con}(\mathsf{BG}(4))$ is indeed true in the standard coordinate structure since $\mathsf{BG}(4)$ is consistent (for it has a model). This sentence becomes provable when further set axioms are added. On the other hand, $\mathsf{BG}(4)$ has a little brother, $\mathsf{BG}_0(4)$, which is a first-order theory (we replace the Continuity Axiom by infinitely many instances of the Continuity axiom scheme). Then, conservativeness holds for $\mathsf{BG}_0(4)$ because it is complete! (As we know from the aforementioned celebrated result by Tarski 1951.) But now the required representation theorem does not hold for the little brother $\mathsf{BG}_0(4)$. Instead, a rather different representation theorem holds, replacing $\mathbb{R}^{n}$ by $F^{n}$ for “some real-closed field $F$.” This is a revision of theoretical physics, for physics works with a manifold, a point set equipped with a system of charts, which are maps into $\mathbb{R}^{n}$. Field’s program required both conservativeness (to vindicate the claimed “instrumentalist nature” of mathematics) and representation (to vindicate the claimed “purely representational” feature of applied mathematics). But the technical snag is that we cannot have both conservativeness and the representation theorem.

4 Main Results About $\mathsf{Gal}(1,3)$

4.1 Definitions: Betweenness Geometry

Definition 15. The formula $\texttt{Bet}(p,q,r)\vee\texttt{Bet}(q,r,p)\vee\texttt{Bet}(r,p,q)$ expresses that points $p,q,r$ are collinear. Assuming $p\neq q$, we use $\ell(p,q)$ to mean the set of points collinear with $p$ and $q$, i.e., the line through $p,q$. It can be proved in $\mathsf{BG}(4)$ that each line is determined by exactly two points. We may express notions of coplanarity, cohyperplanarity, and so on through all positive integer dimensions using formulas that I write as $\texttt{co}_{n}(p_{1},\dots, p_{n+2})$.³⁴ So, $\texttt{co}_{1}(p,q,r)$ means that $p,q,r$ are collinear; $\texttt{co}_{2}(p,q,r,s)$ means that $p,q,r,s$ are coplanar; and so on through higher dimensions. Lines $\ell(p,q)$ and $\ell(r,s)$ are parallel if and only if $\texttt{co}_{2}(p,q,r,s)$ and either $\ell(p,q)=\ell(r,s)$, or $\ell(p,q)$ and $\ell(r,s)$ do not intersect (i.e., have no point in common). For this, we write $\ell(p,q)\parallel\ell(r,s)$. Four distinct points $p,q,r,s$ form a parallelogram just if $\ell(p,q)\parallel\ell(r,s)$ and $\ell(p,s)\parallel\ell(q,r)$ (see Bennett 1995, 49). The notion of what I call a 4-frame is given below (definition 58, in appendix 5.2): an ordered quintuple $O,X,Y,Z,I$ that do not lie in the same 3-dimensional space.

The theory $\mathsf{BG}(4)$ proves the existence of a 4-frame: this is simply the Lower Dimension Axiom (the axioms are listed in appendix 5.1). It can be proved in $\mathsf{BG}(4)$ that, given a line $\ell$ and a point $p$, there is a unique line $\ell'$ parallel to $\ell$ and containing $p$ (this is called Playfair’s Axiom and is an equivalent of Euclid’s Parallel Postulate). From Playfair’s Axiom, it can be proved in $\mathsf{BG}(4)$ that $\parallel$ is an equivalence relation. A number of other theorems from plane and solid geometry can be established, including Desargues’s Theorem and Pappus’s Theorem. See Bennett (1995) for an explanation of these theorems. It can be proved that there is a bijection between any pair of lines. The claims mentioned so far are sufficient (the assumptions required include Desargues’s Theorem and Pappus’s Theorem) to establish that, given distinct parameters $p,q$, the line $\ell(p,q)$ is isomorphic to an ordered field.³⁵ The Continuity Axiom of $\mathsf{BG}(4)$ then ensures that this field is order-complete. From this, we conclude that there is a (unique) isomorphism $\varphi_{p,q}:\ell(p,q)\to\mathbb{R}$, i.e., $\varphi_{p,q}(p)=0$ and $\varphi_{p,q}(q)=1$. See also the proof sketch for theorem 62 below.

4.2 Definitions: Galilean Geometry

Turning to the system $\mathsf{Gal}(1,3)$, we need separate definitions of notions pertaining to simultaneity ($\sim$) and sim-congruence ($\equiv^{\sim}$).

Definition 16. A time axis $T$ is a line $\ell(p,q)$, where $p\not\sim q$.

Definition 17. A simultaneity hypersurface $\Sigma_p$ is the set $\{q\mid q\sim p\}$ of points simultaneous with $p$.

Beyond the notion of a 4-frame, we need a few more specialized notions of “frame” for Galilean spacetime.

Definition 18 (sim 4-frame). A sim 4-frame is a sequence of five points $O,X,Y,Z,I$ such that $O,X,Y,Z$ are simultaneous and not coplanar, and $I$ is not simultaneous with $O$. A sim 4-frame is automatically a 4-frame.

Definition 19 (Euclidean sim 3-frame). A Euclidean sim 3-frame is a sequence of four points $O,X,Y,Z$ that are simultaneous, are not $\texttt{co}_2$, and $OX$, $OY$, $OZ$ have the same length and are mutually perpendicular. That is, $OX\equiv^{\sim}OY$, $OX\equiv^{\sim}OZ$, and $OY\equiv^{\sim}OZ$; and $OX\perp^{\sim}OY$, $OX\perp^{\sim}OZ$, and $OY\perp^{\sim}OZ$.³⁶

Definition 20 (Galilean 4-frame). A Galilean 4-frame is a sequence of five points $O,X,Y,Z,I$ that are a sim 4-frame and such that the four points $O,X,Y,Z$ are a Euclidean sim 3-frame. Note that $O\not\sim I$, and then the line $\ell(O,I)$ is called the time axis of the Galilean 4-frame. A Galilean 4-frame is automatically a 4-frame. We shall simply call it a Galilean frame.

4.3 Soundness

It is straightforward to demonstrate that $\mathsf{Gal}(1,3)$ is true in the coordinate structure $\mathbb{G}^{(1,3)}$ by verifying that each axiom of $\mathsf{Gal}(1,3)$ is true in $\mathbb{G}^{(1,3)}$.

Lemma 21 (Soundness Lemma). $\mathbb{G}^{(1,3)}\models_{2}\mathsf{Gal}(1,3)$.

4.4 Lemmas

Lemma 22. Given a point $p$ and a time axis $T$, there is a unique line $\ell'\parallel T$ st $p\in\ell'$. (This is Playfair’s Axiom, a theorem of $\mathsf{BG}(4)$, and an equivalent of Euclid’s parallel postulate.)

Lemma 23. Any five simultaneous points are $\texttt{co}_{3}$ (i.e., $\text{cohyperplanar}_{3}$).

Proof. This follows from the Upper Dimension Axiom in $\mathsf{EG}(3)^{\sim}$. This asserts that, for a fixed simultaneity hypersurface $\Sigma_{z}$, any five points in $\Sigma_{z}$ are $\texttt{co}_{3}$. Hence, any five simultaneous points are $\texttt{co}_{3}$.∎

Lemma 24 (Non-Triviality). There are at least two non-simultaneous points.

Proof. By the Lower Dimension axiom in $\mathsf{BG}(4)$, there is a 4-frame of five points, $O,X,Y,Z,I$, which are not $\texttt{co}_{3}$. By lemma 23, any five simultaneous points are $\texttt{co}_{3}$. If $O\sim X\sim Y\sim Z\sim I$, they’d be $\texttt{co}_{3}$, a contradiction. So, there are at least two non-simultaneous points.∎

Lemma 25 (Galilean Frame Lemma). There is a Galilean frame $O,X,Y,Z,I$.

Proof. By lemma 24, let $O,I$ be two non-simultaneous points. By $\mathsf{EG}(3)^{\sim}$, Euclidean three-dimensional geometry holds on simultaneity hypersurface $\Sigma_{O}$. So, there exists $O,X,Y,Z$, a Euclidean sim 3-frame in $\Sigma_{O}$. Since $O$ and $I$ are not simultaneous, $O,X,Y,Z,I$ form a Galilean frame (whose time axis is $\ell(O,I)$).∎

4.5 Vector Methods

In the first part of this section, we first assume that we are considering a full model $M\models_{2}\mathsf{BG}(4)$, with $M=(\mathbb{P},B)$ (i.e., $B\subseteq\mathbb{P}^{3}$ is the interpretation in $M$ of the predicate $\texttt{Bet}$). And then, we further assume we are considering a full model $M\models_{2}\mathsf{Gal}(1,3)$, with $M=(\mathbb{P},B,\sim,\equiv^{\sim})$. We assume the material in appendix 5.4, which introduces the new sorts: reals and vectors.³⁷ The vector displacement from $p$ to $q$ is written: $\mathbf{v}_{p,q}$.³⁸ In particular, recall that, by theorem 68, the vector space $\mathbb{V}$ of displacements is isomorphic to $\mathbb{R}^{4}$ (as a vector space).³⁹

Since $M\models_{2}\mathsf{BG}(4)$, we know, by theorem 62, that there exists a coordinate system $\Phi:\mathbb{P}\to\mathbb{R}^{4}$ on $M$, i.e., an isomorphism $\Phi:(\mathbb{P},B)\to(\mathbb{R}^{4},B_{\mathbb{R}^{4}})$.

Definition 26. Let $O, X,Y,Z,I$ be a 4-frame in $M$. Define the four vectors:

\[\begin{aligned} \mathbf{e}_{1}:=\mathbf{v}_{O,X};\quad \mathbf{e}_{2}:=\mathbf{v}_{O,Y};\quad \mathbf{e}_{3}:=\mathbf{v}_{O,Z};\quad \mathbf{e}_{4}:=\mathbf{v}_{O,I}. \end{aligned}\qquad{(25)}\]

Lemma 27. $\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}\}$ is a basis for $\mathbb{V}$.

This is established inside the detailed proof of theorem 68 below.

Definition 28. Given a coordinate system $\Phi$ on $M$, we can define the associated 4-frame, $O,X,Y,Z,I$ of points in $M$:

\[\begin{gathered} O:=\Phi^{-1}(\mathbf{O}),\quad X:=\Phi^{-1}(\mathbf{X}),\quad Y:=\Phi^{-1}(\mathbf{Y}),\\ Z:=\Phi^{-1}(\mathbf{Z}),\quad I:=\Phi^{-1}(\mathbf{I}). \end{gathered}\qquad{(26)}\]

Definition 29. Given a coordinate system $\Phi$, we define four basis vectors:

\[\begin{aligned} \mathbf{e}^{\Phi}_{1}:=\mathbf{v}_{O,X};\quad \mathbf{e}^{\Phi}_{2}:=\mathbf{v}_{O,Y};\quad \mathbf{e}^{\Phi}_{3}:=\mathbf{v}_{O,Z};\quad \mathbf{e}^{\Phi}_{4}:=\mathbf{v}_{O,I}. \end{aligned}\qquad{(27)}\]

Lemma 30. $\{\mathbf{e}^{\Phi}_{1},\mathbf{e}^{\Phi}_{2},\mathbf{e}^{\Phi}_{3},\mathbf{e}^{\Phi}_{4}\}$ is a basis for $\mathbb{V}$.

This is a corollary of lemma 27.

Given a coordinate system $\Phi$ and a point $p$, the four components of $\Phi(p)$ are written as follows:

\[\Phi(p)=\begin{pmatrix}\Phi^{1}(p)\\\Phi^{2}(p)\\\Phi^{3}(p)\\\Phi^{4}(p)\end{pmatrix}.\qquad{(28)}\]

Lemma 31. For any point $p$, we have:

\[\mathbf{v}_{O,p}=\sum_{a=1}^{4}\Phi^{a}(p)\,\mathbf{e}^{\Phi}_{a}.\qquad{(29)}\]

Proof. Consider some of the details of the proof of the Representation Theorem for $\mathsf{BG}(4)$ (see Theorem 62 below). Examining the vector $\mathbf{v}_{O,p}$ from the origin $O$ to $p$, one can see that:

\[\mathbf{v}_{O,p}=\mathbf{v}_{O,p_{X}}+\mathbf{v}_{O,p_{Y}}+ \mathbf{v}_{O,p_{Z}}+\mathbf{v}_{O,p_{I}},\qquad{(30)}\]

where $p_{X}$, $p_{Y}$, $p_{Z}$, and $p_{I}$ are the “ordinates” on the four axes. Note first that $\mathbf{v}_{O,p_{X}}=\varphi_{O,X}(p_{X})\mathbf{v}_{O,{X}}=\varphi_{O,X}(p_{X})\mathbf{e}^{\Phi}_{1}$, and similarly for the other three vectors. So:

\[\mathbf{v}_{O,p}=\varphi_{O,X}(p_{X})\mathbf{e}^{\Phi}_{1}+ \varphi_{O,Y}(p_{Y})\mathbf{e}^{\Phi}_{2}+\varphi_{O,Z}(p_{Z}) \mathbf{e}^{\Phi}_{3}+\varphi_{O,I}(p_{I})\mathbf{e}^{\Phi}_{4}.\qquad{(31)}\]

Note second that $\Phi^{1}(p)$ is defined to be $\varphi_{O,X}(p_X)$, and $\Phi^{2}(p)$ is defined to be $\varphi_{O,Y}(p_Y)$, and similarly for $Z$ and $I$. Hence:

\[\mathbf{v}_{O,p}=\Phi^{1}(p)\mathbf{e}^{\Phi}_{1}+\Phi^{2}(p) \mathbf{e}^{\Phi}_{2}+\Phi^{3}(p)\mathbf{e}^{\Phi}_{3}+\Phi^{4}(p) \mathbf{e}^{\Phi}_{4}.\qquad{(32)}\]∎

Lemma 32. $\mathbf{v}_{p,q}=\sum_{a=1}^{4}(\Phi^a(q)-\Phi^{a}(p))\,\mathbf{e}^{\Phi}_{a}$.

Proof. This is verified as follows:

\[\begin{aligned} \mathbf{v}_{p,q} & =\mathbf{v}_{p,O}+\mathbf{v}_{O,q}=(-\mathbf{v}_{O,p})+\mathbf{v}_{O,q}=\mathbf{v}_{O,q}-\mathbf{v}_{O,p}\\ & =\sum_{a=1}^{4}\Phi^{a}(q)\,\mathbf{e}^{\Phi}_{a}-\sum_{a=1}^{4}\Phi^{a}(p)\,\mathbf{e}^{\Phi}_{a}\\ & =\sum_{a=1}^{4}(\Phi^{a}(q)-\Phi^{a}(p))\,\mathbf{e}^{\Phi}_{a},\end{aligned}\qquad{(33)}\]

where we used Chasles’s Relation (i.e., $\mathbf{v}_{p,q}+\mathbf{v}_{q,r}=\mathbf{v}_{p,r}$), some properties of vectors, and then lemma 31 to expand $\mathbf{v}_{O,q}$ and $\mathbf{v}_{O,p}$ into their components in the $\Phi$-basis.∎

Note that the vector $\mathbf{v}_{p,q}$ from $p$ to $q$ is entirely coordinate-independent.

Let us now assume we are considering a full model $M\models_{2}\mathsf{Gal}(1,3)$, with $M=(\mathbb{P},B,\sim,\equiv^{\sim})$.

Lemma 33. Any simultaneity hypersurface $\Sigma$ in $M$ is a three-dimensional affine space.

Proof. If $\Sigma_{p}$ is a simultaneity hypersurface, then, by $\mathsf{EG}(3)^{\sim}$, the restriction $(\Sigma_{p},B\upharpoonright_{\Sigma_{p}},(\equiv^{\sim})\upharpoonright_{\Sigma_{p}})$ is a Euclidean three-space isomorphic to $(\mathbb{R}^{3},B_{\mathbb{R}^{3}},\equiv_{\mathbb{R}^{3}})$ by theorem 63. Since the reduct $(\Sigma_{p},B\upharpoonright_{\Sigma_{p}})$ (i.e., forgetting the congruence relation) of a Euclidean 3-space is an affine 3-space, $\Sigma_{p}$ is an affine three-space and indeed isomorphic to $(\mathbb{R}^{3},B_{\mathbb{R}^{3}})$.∎

Definition 34. We define the horizontal, or simultaneity, vector subspace $\mathbb{V}^{\sim}$ as follows:

\[\mathbb{V}^{\sim}:=\{\mathbf{v}_{p,q}\in\mathbb{V}\mid p\sim q\}.\qquad{(34)}\]

Definition 34 yields:

Lemma 35. $p\sim q$ iff $\mathbf{v}_{p,q}\in\mathbb{V}^{\sim}$.

From lemma 33, we obtain:

Lemma 36. $\mathbb{V}^{\sim}$ is a three-dimensional linear subspace of $\mathbb{V}$.

Definition 37. We define $p+\mathbb{V}^{\sim}:=\{q\in\mathbb{P}\mid\mathbf{v}_{p,q}\in\mathbb{V}^{\sim}\}$.

Lemma 38. $q\in p+\mathbb{V}^{\sim}$ if and only if $p\sim q$.

Proof. This is immediate from definition 37 and lemma 35.∎

Lemma 39. $\Sigma_{p}=p+\mathbb{V}^{\sim}$.

Proof. $q\in\Sigma_{p}$, if and only if $p\sim q$, if and only if (lemma 38) $q\in p+\mathbb{V}^{\sim}$.∎

Lemma 40. Let a Galilean frame $O,X,Y,Z,I$ be given, and let $\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}$ be defined as in definition 26. Then the subset $\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}$ is a basis for $\mathbb{V}^{\sim}$.

Proof. The proof is that the vectors $\mathbf{v}_{O,X}$, $\mathbf{v}_{O,Y}$, and $\mathbf{v}_{O,Z}$ each lie in $\mathbb{V}^{\sim}$, and, moreover, given any point $q\in\Sigma_{O}$, the vector $\mathbf{v}_{O,q}$ is a linear combination of $\mathbf{v}_{O,X}$, $\mathbf{v}_{O,Y}$, and $\mathbf{v}_{O,Z}$.∎

Lemma 41. Given a coordinate system $\Phi$, the set $\{\mathbf{e}^{\Phi}_{1},\mathbf{e}^{\Phi}_{2},\mathbf{e}^{\Phi}_{3}\}$ is a basis for $\mathbb{V}^{\sim}$.

This is a corollary of the previous lemma.

Lemma 42. Let a Galilean frame $O,X,Y,Z,I$ be given, and let $\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}$ be defined as in definition 26 above. Let $\mathbf{v}\in\mathbb{V}$ with $\mathbf{v}=\sum_{i=1}^{4}v^{i}\mathbf{e}_{i}$. Then

\[\mathbf{v}\in\mathbb{V}^{\sim}\leftrightarrow v^{4}=0.\qquad{(35)}\]

Proof. Let $p$ be any point, and consider:

\[p'=p+\mathbf{v}=p+\sum_{i=1}^{3}v^{3}\mathbf{e}_{i}+\sum_{i=1}^{4}v^{4}\mathbf{v}_{O,I}.\qquad{(36)}\]

So, $\mathbf{v}=\mathbf{v}_{p,p'}$. If $\mathbf{v}_{p,p'}\in\mathbb{V}^{\sim}$, we infer that: $\mathbf{v}_{p,p'}=\alpha^{1}\mathbf{e}_{1}+\alpha^{2}\mathbf{e}_{2}+\alpha^{3}\mathbf{e}_{3}$ (for some coefficients $\alpha^{i}\in\mathbb{R}$) by lemma 36. Equating coefficients, we conclude that $\alpha^{i}=v^{i}$ (for $i=1,2,3$) and $v^{4}=0$, as claimed. Conversely, if $v^{4}=0$, we infer: $\mathbf{v}_{p,p'}=\sum_{i=1}^{3}v^{i}\mathbf{e}_{i}+\sum_{i=1}^{4}0.\mathbf{v}_{O,I}=\sum_{i=1}^{3}v^{i}\mathbf{e}_{i}$. And thus, $\mathbf{v}_{p,p'}\in\mathbb{V}^{\sim}$. This implies that $\mathbf{v}\in\mathbb{V}^{\sim}$.∎

Definition 43. Let $\Sigma_{p}$ and $\Sigma_{q}$ be simultaneity hypersurfaces. We say that $\Sigma_{p}$ is parallel to $\Sigma_{q}$ if and only if either $\Sigma_{p}=\Sigma_{q}$ or there is no intersection of $\Sigma_{p}$ and $\Sigma_{q}$. This is written: $\Sigma_{p}\parallel\Sigma_{q}$.

Lemma 44. All simultaneity hypersurfaces are parallel.

Proof. Let $\Sigma_{p}$ and $\Sigma_{q}$ be simultaneity hypersurfaces. For a contradiction, suppose $\Sigma_{p}\not\parallel\Sigma_{q}$. So, $\Sigma_{p}\neq\Sigma_{q}$, and there is an intersection $r\in\Sigma_{p}\cap\Sigma_{q}$. So, $r\sim p$ and $r\sim q$. Hence, $p\sim q$. Hence, $\Sigma_{p}=\Sigma_{q}$, a contradiction.∎

Lemma 45 (Translation Invariance of Simultaneity). If $p\sim q$, then $(p+\mathbf{v})\sim (q+\mathbf{v})$.

Proof. Suppose $p\sim q$. So, we have: $\mathbf{v}_{p,q}\in\mathbb{V}^{\sim}$. Consider $p'=p+\mathbf{v}$ and $q'=q+\mathbf{v}$. Let $\mathbf{w}=\mathbf{v}_{p,q}$. Since $q=p+\mathbf{w}$, we have $q+\mathbf{v}=(p+\mathbf{w})+\mathbf{v}$, which implies (using some properties of vector addition and the action) $q'=p'+\mathbf{w}$. Hence, $\mathbf{w}=\mathbf{v}_{p',q'}$. So, $\mathbf{v}_{p',q'}=\mathbf{v}_{p,q}$. Since $\mathbf{v}_{p,q}\in\mathbb{V}^{\sim}$, we infer: $\mathbf{v}_{p',q'}\in\mathbb{V}^{\sim}$. From this, it follows that $p'\sim q'$.∎

Lemma 46. Given a simultaneity hypersurface $\Sigma$ and time axis $T$, there is a unique intersection point lying in both $\Sigma$ and $T$.

Proof. Let hypersurface $\Sigma$ and time axis $T$ be given. There cannot be two distinct intersections, say $q$ and $q'$, for then we should have $q\sim q'$, contradicting the assumption that $T$ is a time axis. To establish the existence of at least one intersection, let us fix a Galilean frame $O,X,Y,Z,I$ with $O,I\in T$, i.e., $T=\ell(O,I)$. For any point $p$, we have that there exist unique coefficients $v^{i}$ and $v^{4}$ such that:

\[p=O+\sum_{i=1}^{3}v^{i}\mathbf{e}_{i}+v^{4}\mathbf{v}_{O,I}.\qquad{(37)}\]

Pick any point $p\in\Sigma$ (so $\Sigma=\Sigma_{p}$). Next, define the point $q$:

\[q:=O+v^{4}\mathbf{v}_{O,I}.\qquad{(38)}\]

Then, we infer $\mathbf{v}_{O,q}=v^{4}\mathbf{v}_{O,I}$, which implies that $q\in T$. Next, consider $\mathbf{v}_{q,p}$:

\[\mathbf{v}_{q,p}=\mathbf{v}_{q,O}+\mathbf{v}_{O,p}=-v^{4}\mathbf{v}_{O,I}+\sum_{i=1}^{3}v^{i}\mathbf{e}_{i}+v^{4}\mathbf{v}_{O,I}=\sum_{i=1}^{3}v^{i}\mathbf{e}_{i}.\qquad{(39)}\]

Since $\mathbf{v}_{q,p}=\sum_{i=1}^{3}v^{i}\mathbf{e}_{i}$ and $\sum_{i=1}^{3}v^{i}\mathbf{e}_{i}\in\mathbb{V}^{\sim}$, it follows that $q\sim p$. This implies that $q\in\Sigma_{p}$, and therefore $q\in\Sigma$. The defined point $q$ is, therefore, the required intersection of $T$ and $\Sigma$.∎

Definition 47. Let $\ell=\ell(p,q)$ (with $p\neq q$) be a line, and let $\Sigma$ be a simultaneity hypersurface. We say that $\ell$ is parallel to $\Sigma$ if and only if either $\ell\subseteq\Sigma$ or there is no intersection $r\in T\cap\Sigma$. This is written: $\ell\parallel\Sigma$.

Lemma 48. No time axis is parallel to a simultaneity hypersurface.

Proof. Let $T=\ell(p,q)$ be a time axis (i.e., $p\not\sim q$). Let $\Sigma$ be a simultaneity hypersurface. For a contradiction, suppose $T\parallel\Sigma$. So, either $\ell(p,q)\subseteq\Sigma$ or there is no intersection $r\in T\cap\Sigma$. But, by lemma 46, there is a unique intersection $r\in T\cap\Sigma$. So, we must have: $\ell(p,q)\subseteq\Sigma$. Then, since $p,q\in\ell(p,q)$, we have $p,q\in\Sigma$. Hence, $p\sim q$, a contradiction. Therefore, $T\not\parallel\Sigma$.∎

Lemma 49. Let lines $\ell(p,q)$ and $\ell(r,s)$ be parallel. Then, for some $\alpha\neq 0$, $\mathbf{v}_{p,q}=\alpha\mathbf{v}_{r,s}$.

Proof. This follows from the detailed construction of $\mathbb{V}$ (based on parallelograms and equipollence), which yields theorem 68.∎

Lemma 50. Any line parallel to a time axis is a time axis.

Proof. Suppose line $\ell(p,q)$ is parallel to a time axis $T=\ell(O,I)$, with $O\not\sim I$. Then, by lemma 49, $\mathbf{v}_{p,q}=\alpha\mathbf{v}_{O,I}$, with $\alpha\neq 0$. Since $O\not\sim I$, we have $\mathbf{v}_{O,I}\notin\mathbb{V}^{\sim}$. In general, for any $\alpha\neq 0$, $\mathbf{v}\in\mathbb{V}^{\sim}$ if and only if $\alpha\mathbf{v}\in\mathbb{V}^{\sim}$. So, it follows that $\mathbf{v}_{p,q}\notin\mathbb{V}^{\sim}$. Hence, $p\not\sim q$. Thus, $\ell(p,q)$ is a time axis.∎

4.6 Representation

Definition 51. Let $M=(\mathbb{P},B,\sim,\equiv^{\sim})$ be a $\sigma_{\mathsf{Gal}}$-structure (i.e., $B$ interprets $\texttt{Bet}$, $\sim$ interprets $\sim$, and $\equiv^{\sim}$ interprets $\equiv^{\sim}$). Suppose that $M\models_{2}\mathsf{Gal}(1,3)$. Let $\Phi:\mathbb{P}\to\mathbb{R}^{4}$ be a function. We say:

$B_{\mathbb{R}^{4}}$ represents $B$ wrt $\Phi$	iff	for all $p,q,r\in\mathbb{P}$: $B(p,q,r) \leftrightarrow(\Phi(p),\Phi(q),\Phi(r))\in B_{\mathbb{R}^{4}}$.
$\sim_{\mathbb{R}^{4}}$ represents $\sim$ wrt $\Phi$	iff	for all $p,q\in\mathbb{P}$: $p\sim q \leftrightarrow\Phi(p)\sim_{\mathbb{R}^{4}}\Phi(q)$.
$\equiv^{\sim}_{\mathbb{R}^{4}}$ represents $\equiv^{\sim}$ wrt $\Phi$	iff	for all $p,q,r,s\in\mathbb{P}$: $pq\equiv^{\sim}rs \leftrightarrow\Phi(p)\Phi(q)\equiv^{\sim}_{\mathbb{R}^{4}} \Phi(r)\Phi(s)$.

If $\Phi$ is a bijection and each of the three above representation conditions holds, then $\Phi$ is an isomorphism from $M$ to $\mathbb{G}^{(1,3)}$.

In order to prove the Representation Theorem for $\mathsf{Gal}(1,3)$, we need to establish three main lemmas. I call these the Chronology Lemma, the Galilean Frame Translation Invariance Lemma, and the Congruence Lemma.

4.7 The Chronology Lemma

Lemma 52 (Chronology). Let $M=(\mathbb{P},B,\sim,\equiv^{\sim})$ be a $\sigma_{\mathsf{Gal}}$-structure, with $M\models_{2}\mathsf{Gal}(1,3)$. Let $O,X,Y,Z,I$ be a sim 4-frame in $M$. Since $(\mathbb{P},B)\models_{2}\mathsf{BG}(4)$, let $\Phi:(\mathbb{P},B)\to(\mathbb{R}^{4},B_{\mathbb{R}^{4}})$ be an isomorphism matching $O,X,Y,Z,I$. Then $\sim_{\mathbb{R}^{4}}$ represents $\sim$ wrt $\Phi$.

Proof. Since $O,X,Y,Z,I$ is a sim 4-frame, the points $O,X,Y,Z$ are simultaneous, not coplanar, and $O\not\sim I$. Given that $\Phi$ matches $O,X,Y,Z,I$, with $O,X,Y,Z$ simultaneous, the associated basis $\{\mathbf{e}^{\Phi}_{1},\mathbf{e}^{\Phi}_{2},\mathbf{e}^{\Phi}_{3}\}$ is a basis for the simultaneity vector space $\mathbb{V}^{\sim}$, by lemma 41. Since a sim 4-frame is a 4-frame, $\{\mathbf{e}^{\Phi}_{1},\mathbf{e}^{\Phi}_{2},\mathbf{e}^{\Phi}_{3},\mathbf{e}^{\Phi}_{4}\}$ is a basis for $\mathbb{V}$. Let points $p,q$ be given. We claim:

\[p\sim q\quad\leftrightarrow\quad\Phi^{4}(p)=\Phi^{4}(q).\qquad{(40)}\]

From lemma 35, we have that $p\sim q$ holds if and only if $\mathbf{v}_{p,q}\in\mathbb{V}^{\sim}$. Using lemma 32, we next expand $\mathbf{v}_{p,q}$ in the basis $\{\mathbf{e}^{\Phi}_a\}$ determined by $\Phi$:

\[\mathbf{v}_{p,q}=\sum_{a=1}^{4}(\Phi^a(q)-\Phi^{a}(p))\mathbf{e}^{\Phi}_{a}.\qquad{(41)}\]

From lemma 42, we conclude that $\mathbf{v}_{p,q}\in\mathbb{V}^{\sim}$ iff $(\mathbf{v}_{p,q})^{4}=0$. That is, $p\sim q$ iff $\Phi^{4}(q)-\Phi^{4}(p)=0$. And therefore, $p\sim q$ iff $\Phi^{4}(q)=\Phi^{4}(p)$, as claimed.∎

4.8 The Galilean Frame Translation Invariance Lemma

Lemma 53 (Galilean Frame Translation Invariance). Let $M=(\mathbb{P},B,\sim,\equiv^{\sim})$ be a $\sigma_{\mathsf{Gal}}$-structure, with $M\models_{2}\mathsf{Gal}(1,3)$. Let $O,X,Y,Z,I$ be a Galilean 4-frame in $M$. Since $(\mathbb{P},B)\models_{2}\mathsf{BG}(4)$, let $\Phi:(\mathbb{P},B)\to(\mathbb{R}^{4},B_{\mathbb{R}^{4}})$ be an isomorphism matching $O,X,Y,Z,I$. Let $\mathbf{v}\in\mathbb{V}$. Let $O'=O+\mathbf{v}$, $X'=X+\mathbf{v}$, $Y'=Y+\mathbf{v}$, $Z'=Z+\mathbf{v}$, $I'=I+\mathbf{v}$. Then $O',X',Y',Z',I'$ is a Galilean 4-frame.

That is, leaving the assumptions as stated, when we apply a translation (given by a vector $\mathbf{v}$) to a Galilean frame, so $O'=O+\mathbf{v}$, etc., the result is also a Galilean frame:

\[O,X,Y,Z,I\text{ is a Galilean 4-frame iff }O',X',Y',Z'\text{ is a Galilean 4-frame.}\qquad{(42)}\]

Proof. Without loss of generality, we may suppose that $\mathbf{v}$ does not lie in the simultaneity hypersurface $\Sigma_{O}$. For if it does, the vector will simply translate the frame “horizontally,” along within $\Sigma_{O}$ and the Euclidean axioms, along with the fact that the temporal benchmark point $I$ also moves “horizontally” too within the hypersurface $\Sigma_{I}$, guarantee that $O',X',Y',Z',I'$ is a Galilean 4-frame.

I will sketch how the proof goes. It is best illustrated by figure 2.

Figure 2: “Transformed Galliean frame” on \Sigma_{O'} (axis \ell(O,Z) and point I' suppressed). — Figure 2: “Transformed Galliean frame” on $\Sigma_{O'}$ (axis $\ell(O,Z)$ and point $I'$ suppressed).

This is the sole part of our analysis appealing to the axiom $\mathsf{Gal}5$, stating the translation invariance of $\equiv^{\sim}$.

The five points $O,X,Y,Z,I$ form a Galilean frame, and thus the four points $O,X,Y,Z$ form a Euclidean sim 3-frame. So, in the lower simultaneity hypersurface, $\Sigma_O$, we have a Euclidean sim 3-frame $O,X,Y,Z$: the three legs $OX$, $OY$, and $OZ$ are perpendicular and of equal length. (The point $Z$ and the axis $\ell(O,Z)$ are suppressed in figure 2.)

Consider the hypersurface $\Sigma_{O'}$. By assumption, each of the points $O',X',Y',Z'$ is obtained by adding the same displacement vector: $\mathbf{v}=\mathbf{v}_{O,O'}$:

\[\begin{gathered} O'=O+\mathbf{v},\quad X'=X+\mathbf{v},\\ Y'=Y+\mathbf{v},\quad Z'=Z+\mathbf{v}. \end{gathered}\qquad{(43)}\]

Since $O,X,Y,Z$ are simultaneous, it follows, using lemma 45, that $O',X',Y',Z'$ are simultaneous. So, all four points lie in $\Sigma_{O'}$.

Next, we use the Translation Invariance axiom $\mathsf{Gal}5$ of $\mathsf{Gal}(1,3)$: $\equiv^{\sim}$ is translation invariant. Since $O,X,Y,Z$ form a Euclidean sim 3-frame, we may conclude, from the translation invariance of $\equiv^{\sim}$, that $O',X',Y',Z'$ is also a Euclidean sim 3-frame. Since $\mathbf{v}$ does not lie parallel to $\Sigma_{O}$, $I'$ is not simultaneous with $O',X',Y',Z'$. And, so, $O',X',Y',Z',I'$ is a Galilean 4-frame.∎

4.9 The Congruence Lemma

Lemma 54 (Congruence). Let $M=(\mathbb{P},B,\sim,\equiv^{\sim})$ be a $\sigma_{\mathsf{Gal}}$-structure, with $M\models_{2}\mathsf{Gal}(1,3)$. Let $O,X,Y,Z,I$ be a Galilean 4-frame in $M$. By the Chronology Lemma (lemma 52), there is an isomorphism $\Phi:(\mathbb{P},B,\sim)\to(\mathbb{R}^{4},B_{\mathbb{R}^{4}},\sim_{\mathbb{R}^{4}})$ matching $O,X,Y,Z,I$. Then, $\equiv^{\sim}_{\mathbb{R}^{4}}$ represents $\equiv^{\sim}$ with respect to $\Phi$.

Proof. We are given a structure $M=(\mathbb{P},B,\sim,\equiv^{\sim})$, a Galilean frame, $O,X,Y,Z,I$ in $M$, and an isomorphism $\Phi:(\mathbb{P},B,\sim)\to(\mathbb{R}^{4},B_{\mathbb{R}^{4}},\sim_{\mathbb{R}^{4}})$, matching $O,X,Y,Z,I$. We shall call $\Phi$ the “global isomorphism.” We claim that $\equiv^{\sim}_{\mathbb{R}^{4}}$ represents $\equiv^{\sim}$ with respect to $\Phi$; that is, for simultaneous points $p,q,r,s$, we have:⁴⁰

\[pq\equiv^{\sim} rs\leftrightarrow\Delta_{3}(\vec{\Phi}(p),\vec{\Phi}(q))=\Delta_3(\vec{\Phi}(r),\vec{\Phi}(s)).\qquad{(44)}\]

Consider figure 3:

Figure 3: “Lifted Euclidean frame” on \Sigma_{u} (axis \ell(O,Z) suppressed). — Figure 3: “Lifted Euclidean frame” on $\Sigma_{u}$ (axis $\ell(O,Z)$ suppressed).

By hypothesis, the five points $O,X,Y,Z,I$ form a Galilean frame, and thus the four points $O,X,Y,Z$ form a Euclidean sim 3-frame. For points in the lower simultaneity hypersurface, $\Sigma_{O}$, we have, from the Euclidean axiom group $\mathsf{EG}(3)^{\sim}$ in $\mathsf{Gal}(1,3)$ and the Representation Theorem for Euclidean geometry (theorem 63), the existence of an isomorphism (i.e., coordinate system on $\Sigma_{O}$),

\[\psi_{O}:(\Sigma_{O},B\upharpoonright_{\Sigma_{O}},(\equiv^{\sim})\upharpoonright_{\Sigma_{O}})\to(\mathbb{R}^{3},B_{\mathbb{R}^{3}},\equiv_{\mathbb{R}^{3}}),\qquad{(45)}\]

that matches this Euclidean sim 3-frame $O,X,Y,Z$. So, in the hypersurface $\Sigma_{O}$, a “mini-representation theorem” holds. For any $p,q,r,s\in\Sigma_{O}$,

\[pq\equiv^{\sim}rs\leftrightarrow\vec{\psi}_{O}(p)\vec{\psi}_{O}(q)\equiv_{\mathbb{R}^{3}}\vec{\psi}_{O}(r),\vec{\psi}_{O}(s).\qquad{(46)}\]

Let $\Phi_{O}$ be $\Phi\upharpoonright_{\Sigma_{O}}$: the restriction of the global isomorphism $\Phi$ to the hypersurface $\Sigma_{O}$. We are also given that $\Phi_{O}$ also matches $O,X,Y,Z$. By the uniqueness of coordinate systems that match the same frame (lemma 66), we conclude:

\[\psi_{O}=\Phi_{O}.\qquad{(47)}\]

Thus, by (46) and (47), $\Phi_{O}$ satisfies:

\[pq\equiv^{\sim}rs\leftrightarrow\vec{\Phi}_{O}(p)\vec{\Phi}_{O}(q)\equiv_{\mathbb{R}^{3}}\vec{\Phi}_{O}(r),\vec{\Phi}_{O}(s).\qquad{(48)}\]

We now repeat the same argument for an arbitrary simultaneity surface, $\Sigma_{u}$.

Given any point $u$, we consider the hypersurface $\Sigma_{u}$. By lemma 46, the time axis $\ell(O,I)$ intersects $\Sigma_{u}$ at the corresponding “origin,” $O_{u}$. By lemma 22, there are unique lines through $X$, $Y$, and $Z$, each parallel to $\ell(O,O_{u})$. By lemma 46 again, these intersect $\Sigma_{u}$ at points $X_{u},Y_{u},Z_{u}$. By lemma 44, the hypersurfaces $\Sigma_{O}$ and $\Sigma_{u}$ are parallel; this guarantees that each of the points $O_{u},X_{u},Y_{u},Z_{u}$ is obtained by adding the same displacement vector: $\mathbf{v}=\mathbf{v}_{O,O_{u}}$:

\[\begin{gathered} O_{u}=O+\mathbf{v},\quad X_{u}=X+\mathbf{v},\\ Y_{u}=Y+\mathbf{v},\quad Z_{u}=Z+\mathbf{v}. \end{gathered}\qquad{(49)}\]

By the Translation Invariance of Galilean frames, lemma 53, since $O,X,Y,Z,I$ form a Galilean 4-frame, we may conclude that $O_{u},X_{u},Y_{u},Z_{u},I_{u}$ (where $I_{u}=I+\mathbf{v}$) also form a Galilean 4-frame. And thus, $O_{u},X_{u},Y_{u},Z_{u}$ form a Euclidean sim 3-frame. By the Representation Theorem for Euclidean geometry, there is an isomorphism $\psi_{u}$, which matches $O_{u},X_{u},Y_{u},Z_{u}$. By similar reasoning to the case of $\Sigma_{O}$, we define the restriction $\Phi_{u}$ to be $\Phi\upharpoonright_{\Sigma_{u}}$—i.e., the restriction of the global isomorphism $\Phi$ to the hypersurface $\Sigma_{u}$. We can conclude:

\[\psi_{u}=\Phi_{u}.\qquad{(50)}\]

Thus, $\Phi_{u}$ satisfies the following: for any points $p,q,r,s\in\Sigma_{u}$,

\[pq\equiv^{\sim}rs\quad\leftrightarrow\quad\vec{\Phi}_{u}(p)\,\vec{\Phi}_{u}(q)\equiv_{\mathbb{R}^{3}}\vec{\Phi}_{u}(r)\,\vec{\Phi}_{u}(s).\qquad{(51)}\]

This is equivalent to (44).∎

5 Representation Theorem for $\mathsf{Gal}(1,3)$

Our main theorem is then the following:

Theorem 55 (Representation Theorem for Galilean Spacetime). Let $M=(\mathbb{P},B,\sim,\equiv^{\sim})$ be a full $\sigma_{\mathsf{Gal}}$-structure. Then

\[M\models_{2}\mathsf{Gal}(1,3)\quad\text{if and only if}\quad\text{there is an isomorphism }\Phi:M\to\mathbb{G}^{(1,3)}.\qquad{(52)}\]

Proof. For the right-to-left direction, suppose there is an isomorphism $\Phi:M\to\mathbb{G}^{(1,3)}$. So, $M\cong\mathbb{G}^{(1,3)}$. By the Soundness Lemma (lemma 21), $\mathbb{G}^{(1,3)}\models_{2}\mathsf{Gal}(1,3)$. Since isomorphic structures satisfy the same sentences, it follows that $M\models_{2}\mathsf{Gal}(1,3)$.

For the converse, let $M\models_{2}\mathsf{Gal}(1,3)$. From the Galilean Frame Lemma (lemma 25), a Galilean frame $O,X,Y,Z,I$ exists. This is a 4-frame. By the Representation Theorem for $\mathsf{BG}(4)$ (theorem 62), we conclude that there is a global isomorphism:

\[\Phi:\mathbb{P}\to\mathbb{R}^{4}\qquad{(53)}\]

such that $\Phi$ matches the frame $O,X,Y,Z,I$, and $\Phi:(\mathbb{P},B)\to(\mathbb{R}^{4},B_{\mathbb{R}^{4}})$ is an isomorphism. So, $B_{\mathbb{R}^{4}}$ represents the betweenness relation $B$ of $M$ with respect to $\Phi$. Recall that the global isomorphism $\Phi$ matches a Galilean frame $O,X,Y,Z,I$. Since $O,X,Y,Z,I$ is a Galilean frame, $O,X,Y,Z,I$ is a sim frame. By the Chronology Lemma (lemma 52), we conclude that the relation $\sim_{\mathbb{R}^{4}}$ represents the simultaneity relation $\sim$ of $M$ with respect to $\Phi$. What is more, again, since $O,X,Y,Z,I$ is a Galilean frame, we can appeal to the Congruence Lemma (lemma 54) and conclude that $\equiv^{\sim}_{\mathbb{R}^{4}}$ represents the sim-congruence relation $\equiv^{\sim}$ of $M$ with respect to $\Phi$.

Assembling this, $\Phi:M\to\mathbb{G}^{(1,3)}$ is an isomorphism, as claimed.∎

Such isomorphisms $\Phi:M\to\mathbb{G}^{(1,3)}$ are inertial charts on Galilean spacetime. They correspond, one-to-one, with Galilean frames. As we have seen, the transformation group between these isomorphisms (or, if you wish, between the Galilean frames) is precisely $\mathcal{G}^{e}(1,3)$—the extended Galilean group.

Appendices

5.1 Axioms

Definition 56. The non-logical axioms of $\mathsf{BG}(4)$ in $L(\sigma_{\mathsf{Gal},\in})$ are the following nine:⁴¹

Table 3: Order axioms for betweenness.
B1	$\texttt{Bet}$-Identity	$\texttt{Bet}(p,q,p)\to p=q$.
B2	$\texttt{Bet}$-Transitivity	$\texttt{Bet}(p,q,r)\wedge\texttt{Bet}(q,r,s)\wedge q\neq r\to\texttt{Bet}(p,q,s)$.
B3	$\texttt{Bet}$-Connectivity	$\texttt{Bet}(p,q,r)\wedge\texttt{Bet}(p,q,r')\wedge p\neq q \to(\texttt{Bet}(p,r,r')\vee\texttt{Bet}(p,r',r))$.
B4	$\texttt{Bet}$-Extension	$\exists p\,(\texttt{Bet}(p,q,r)\wedge p\neq q)$.

B5	Pasch	$\texttt{Bet}(p,q,r)\wedge\texttt{Bet}(s,u,q) \to\exists x\,(\texttt{Bet}(r, x, s)\wedge\texttt{Bet}(p,u,x))$.
B6	Euclid	$\texttt{Bet}(a,d,t)\wedge\texttt{Bet}(b,d,c)\wedge a\neq d \to\exists x\,\exists y\,(\texttt{Bet}(a,b,x)\wedge \texttt{Bet}(a,c,y)\wedge\texttt{Bet}(x,t,y))$.

B7	Lower Dimension	There exist five points which are not $\texttt{co}_{3}$.
B8	Upper Dimension	Any six points are $\texttt{co}_{4}$.

B9	Continuity Axiom	$[\exists r\,(\forall p\in\mathsf{X}_{1})\,(\forall q\in\mathsf{X}_{2})\, \texttt{Bet}(r,p,q)]\to\exists s\,(\forall p\in\mathsf{X}_{1}) \, (\forall q\in\mathsf{X}_{2})\,\texttt{Bet}(p,s,q)$.

See Szczerba and Tarski (1979, 159–160) for the first-order two-dimensional theory $\textsf{GA}_{2}$ (for “neutral” or “absolute geometry”), which lacks the Euclid Parallel axiom (which is called (E) in Szczerba and Tarski 1979 and is called (Euclid) above). Their system includes Desargues’s Theorem. But, for us, this axiom is no longer required, as it is provable from the remaining axioms in dimensions above two (Szczerba and Tarski 1979, 190). The above axiom system is the second-order, four-dimensional theory and contains (E), i.e., (Euclid). The relevant representation theorem follows from theorem 5.12 of Szczerba and Tarski (1979, 185, see also example 6.1). The same theorem is stated, somewhat indirectly, in Borsuk and Smielew (1960, 196–197). The representation theorem itself goes back to Veblen (1904).

Definition 57. The non-logical axioms of $\mathsf{EG}(3)$ in $L(\sigma_{\texttt{Bet},\equiv,\in})$ are the following eleven:

Table 4: The axioms of Euclidean Geometry for three dimensions.
E1	$\texttt{Bet}$-Identity	$\texttt{Bet}(p,q,p)\to p=q$.
E2	$\equiv$-Identity	$pq\equiv rr\to p=q$.
E3	$\equiv$-Transitivity	$pq\equiv rs\wedge pq\equiv tu\to rs\equiv tu$.
E4	$\equiv$-Reflexivity	$pq\equiv qp$.
E5	$\equiv$-Extension	$\exists r\,(\texttt{Bet}(p,q,r)\wedge qr\equiv su)$.

E6	Pasch	$\texttt{Bet}(p,q,r)\wedge\texttt{Bet}(s,u,r) \to\exists x\,(\texttt{Bet}(q,x,s)\wedge\texttt{Bet}(u,x,p))$.
E7	Euclid	$\texttt{Bet}(a,d,t)\wedge\texttt{Bet}(b,d,c)\wedge a\neq d \to\exists x\,\exists y\,(\texttt{Bet}(a,b,x)\wedge \texttt{Bet}(a,c,y)\wedge\texttt{Bet}(x,t,y))$.
E8	5-Segment	$(p\neq q\wedge\texttt{Bet}(p,q,r)\wedge\texttt{Bet}(p',q',r') \wedge pq\equiv p'q'\wedge qr\equiv q'r'\wedge\ ps\equiv p's'\wedge qs\equiv q's') \to rs\equiv r's'$.

E9	Lower Dimension	There exist four points which are not $\texttt{co}_{2}$.
E10	Upper Dimension	Any five points are $\texttt{co}_{3}$.

E11	Continuity Axiom	$[\exists r\,(\forall p\in\mathsf{X}_{1})\,(\forall q\in\mathsf{X}_{2}) \,\texttt{Bet}(r,p,q)]\to\exists s\,(\forall p\in\mathsf{X}_{1})\, (\forall q\in\mathsf{X}_{2})\,\texttt{Bet}(p,s,q)$.

The original source of this axiomatization is Tarski (1959) and Tarski and Givant (1999). See Tarski (1959, 19–20) for a formulation of the first-order two-dimensional theory, with twelve axioms and one axiom scheme (for continuity); and Tarski and Givant (1999) for a simplification down to ten axioms and one axiom scheme (for continuity). The above axiom system is the second-order, four-dimensional theory (i.e., the single Continuity Axiom is the second-order one).

5.2 Representation Theorems

Definition 58 (4-frame). For betweenness geometry, a 4-frame is an ordered tuple of five points $O,X,Y,Z,I$, which are not $\texttt{co}_{3}$.⁴²

Definition 59 (Perpendicularity). In Euclidean geometry, perpendicularity $OX\perp OY$ for three distinct points $O,X,Y$ is defined as follows: $OX\perp OY$ holds iff $XY\equiv(-X)Y$, where $(-X)$ is the unique point $p$ on $\ell(O,X)$ such that $p\neq X$ and $Op\equiv OX$.

Definition 60 (Euclidean 3-frame). For Euclidean geometry, a Euclidean 3-frame is an ordered quadruple $O,X,Y,Z$ of points that are not $\texttt{co}_{2}$ (i.e., not coplanar) and such that the segments $OX,OY,OZ$ are mutually perpendicular and of equal length.

Definition 61 (Matching). Suppose that $M=(\mathbb{P},B)$ is a $\sigma_{\texttt{Bet}}$-structure with $M\models_{2}\mathsf{BG}(4)$, and suppose that $O,X,Y,Z,I$ is a 4-frame in $M$. Suppose that $\Phi:\mathbb{P}\to\mathbb{R}^{4}$ is a function. We say that $\Phi$ matches $O,X,Y,Z,I$ just if:⁴³

\[\Phi(O)=\mathbf{O},\quad \Phi(X)=\mathbf{X},\quad \Phi(Y)=\mathbf{Y},\quad \Phi(Z)=\mathbf{Z},\quad \Phi(I)=\mathbf{I}.\qquad{(54)}\]

The following two theorems are primarily due to Hilbert (1899), Veblen (1904), and Tarski (1959):⁴⁴

Theorem 62 (Representation Theorem for $\mathsf{BG}(4)$). Let $M=(\mathbb{P},B)$ be a $\sigma_{\texttt{Bet}}$-structure. Assume that $M\models_{2}\mathsf{BG}(4)$. Suppose $O,X,Y,Z,I$ is a 4-frame in $M$. Then there exists a bijection $\Phi:\mathbb{P}\to\mathbb{R}^{4}$ such that:

$\Phi$ matches $O,X,Y,Z,I$.
For all $p,q,r\in\mathbb{P}$: $(p,q,r)\in B\leftrightarrow B_{\mathbb{R}^{4}}(\Phi(p),\Phi(q),\Phi(r))$.

Proof. I give a brief sketch. Given a 4-frame $O,X,Y,Z,I$ in $M$, we first define four lines $\ell(O,X),\ell(O,Y),\ell(O,Z)$, and $\ell(O,I)$: these are the “$x$-axis,” “$y$-axis,” “$z$-axis,” and “$t$-axis” of the 4-frame. One can define (as in Hilbert 1899) geometrical operations $+$, $\times$, and a linear order $\leq$ on each axis (relative to the two fixed parameters that determined that axis). These definitions are explained very clearly in Bennett (1995): for $+$ at p. 48 and for $\times$ at p. 62. Also, see Goldblatt (1987, 23–27). The definition of $\leq$ is given in Tarski (1959, proof of theorem 1). Then, using the betweenness axioms, one shows that, on each axis, $\ell(O,X),\ell(O,Y),\ell(O,Z)$, and $\ell(O,I)$, these definitions specify an ordered field. For details (ignoring the order aspect), see Bennett (1995, 48–72, especially theorem 1, p. 72). What is more, the Continuity Axiom guarantees that this ordered field is a complete ordered field. Up to isomorphism, there is exactly one complete ordered field, and this is also rigid. Consequently, there is a (unique) isomorphism $\varphi_{O,X}:\ell(O,X)\to\mathbb{R}$ (and similarly on each axis):

Figure 5: Isomorphism from \ell(O,X) to \mathbb{R}. — Figure 5: Isomorphism from $\ell(O,X)$ to $\mathbb{R}$.

Given any point $p$, one then constructs four “ordinates” $p_{X},p_{Y},p_{Z},p_{I}$ on the four axes $\ell(O,X),\ell(O,Y),\ell(O,Z),\ell(O,I)$ by certain parallel lines to these axes. Then, one defines the coordinate system $\Phi$ as follows. Given any point $p\in\mathbb{P}$, define:

\[\Phi(p):=\begin{pmatrix}\varphi_{O,X}(p_{X})\\ \varphi_{O,Y}(p_{Y})\\ \varphi_{O,Z}(p_{Z})\\ \varphi_{O,I}(p_{I})\end{pmatrix}.\qquad{(55)}\]

It is clear that $\Phi$ matches $O,X,Y,Z,I$. Finally, one shows that $\Phi$ is a bijection and that it satisfies the required isomorphism condition. Namely, for $p,q,r\in\mathbb{P}$: $B(p,q,r)$ iff $B_{\mathbb{R}^{4}}(\Phi(p),\Phi(q),\Phi(r))$.∎

Theorem 63 (Representation Theorem for $\mathsf{EG}(3)$). Let $M=(\mathbb{P},B,\equiv)$ be a $\sigma_{\texttt{Bet},\equiv}$-structure. Assume that $M\models_{2}\mathsf{EG}(3)$. Suppose $O,X,Y,Z$ is a Euclidean 3-frame in $M$. Then there exists a bijection $\Phi:\mathbb{P}\to\mathbb{R}^{3}$ such that:

(a) $\Phi$ matches $O,X,Y,Z$.

(b) For all $p,q,r\in\mathbb{P}$: $(p,q,r)\in B\leftrightarrow B_{\mathbb{R}^{3}}(\Phi(p),\Phi(q),\Phi(r))$.

Roughly, this corresponds to theorem 1 of Tarski (1959), and a sketch of the proof is given there. The difference is that Tarski considers the two-dimensional first-order theory, whose axioms are what we’ve called $\mathsf{EG}_{0}(2)$, with the first-order continuity axiom scheme. The Representation Theorem in Tarski (1959) asserts that, given a model $M\models\mathsf{EG}_{0}(2)$ and a Euclidean frame, there is a real-closed field $F$ such that the conditions (a), (b), (c) hold, with $\mathbb{R}$ replaced by that field and “$3$” replaced by “$2$.” When we strengthen to the second-order Continuity axiom, it follows that this field is in fact $\mathbb{R}$.

5.3 Automorphisms and Coordinate Systems

Theorem 64. The automorphism (symmetry) groups of the structures defined in definitions 1 and 4 are characterized in table 5.

Table 5: The automorphism groups of standard Euclidean metric space, where $h:\mathbb{R}^{n}\to\mathbb{R}^{n}$.
Aut. Group	Condition
$h\in\text{Aut}(\mathbb{BG}^{n})$	$(\exists A\in GL(n))\,(\exists\mathbf{d}\in\mathbb{R}^{n}) \,(\forall\mathbf{x}\in\mathbb{R}^{n})\ [h(\mathbf{x})=A\mathbf{x}+\mathbf{d}]$
$h\in\text{Aut}(\mathbb{EG}^{n})$	$(\exists R\in O(n))\,(\exists\lambda\in\mathbb{R}-\{0\}) (\exists\mathbf{d}\in\mathbb{R}^{n})\,(\forall\mathbf{x}\in\mathbb{R}^{n}) \ [h(\mathbf{x})=\lambda R\mathbf{x}+\mathbf{d}]$
$h\in\text{Aut} (\mathbb{EG}_{\text{metric}}^{n})$	$(\exists R\in O(n))\,(\exists\mathbf{d}\in\mathbb{R}^{n})\, (\forall\mathbf{x}\in\mathbb{R}^{n})\ [h(\mathbf{x})=R\mathbf{x}+\mathbf{d}]$

Proof. I give a brief summary. For the first, the proof relies on the requirement that straight lines get mapped to straight lines and parallel lines get mapped to parallel lines. The outcome is that any such mapping $h$ must be an affine transformation generated by a $GL(n)$ matrix $A$ and a translation $\mathbf{d}$. So, the automorphism group is what is usually called $\text{Aff}(n)$, the affine group in $n$ dimensions. For the third, the symmetry group is the isometry group of the metric space $\mathbb{EG}_{\text{metric}}^{n}$—thus, what’s usually called the Euclidean group $E(n)$: rotations, inversions, reflections, and translations (reflections and inversions are $O(n)$ matrices with determinant $-1$). For the second, which is less familiar, the symmetries include rotations, inversions, reflections, and translations again, but also include scalings too:

\[\mathbf{x}\mapsto\lambda\mathbf{x}.\qquad{(56)}\]

The latter are sometimes called similitudes or dilations (the non-zero factor $\lambda$ represents this scaling). Although the metric distance between two points is not invariant, nonetheless metric equalities are invariant. Imagine a rubber sheet pinned at some central point, say, $O$, and imagine “stretching” it uniformly and radially from $O$ by some factor. The distance between two points on the sheet is not invariant under the stretching: $\Delta(\mathbf{x},\mathbf{y})\mapsto|\lambda|\Delta(\mathbf{x},\mathbf{y})$, but equality between distances of points (i.e., congruence) is invariant.∎

Lemma 65 (Coordinate Transformations). Given two coordinate systems $\Phi,\Psi:\mathbb{P}\to\mathbb{R}^{4}$, on a full model $M=(\mathbb{P},B)$ of $\mathsf{BG}(4)$, they are related as follows: there is a $GL(4)$ matrix $A$ and a translation $\mathbf{d}\in\mathbb{R}^{4}$ such that, for any point $p\in\mathbb{P}$, we have:

\[\Psi(p)=A\Phi(p)+\mathbf{d}.\qquad{(57)}\]

This follows from two facts. First, if $\Phi,\Psi:M\to(\mathbb{R}^{4},B_{\mathbb{R}^{4}})$ are isomorphisms, then $\Psi\circ\Phi^{-1}\in\text{Aut}((\mathbb{R}^{4},B_{\mathbb{R}^{4}}))$. Second, we have $\text{Aut}((\mathbb{R}^{4},B_{\mathbb{R}^{4}}))=\text{Aff}(4)$. (This is the result given in theorem 64 for the automorphisms of the standard coordinate structure $(\mathbb{R}^{4},B_{\mathbb{R}^{4}})$ for $\mathsf{BG}(4)$.)

Lemma 66. Given a 4-frame $O,X,Y,Z,I$ and two coordinate systems, $\Phi,\Psi$, on a model $M$ of $\mathsf{BG}(4)$, both of which match the frame $O,X,Y,Z,I$, we have:

\[\Psi=\Phi.\qquad{(58)}\]

The proof applies the coordinate transformation equation (57) to the five points, $O,X,Y,Z,I$, which gives five specific instances. The first of these implies that $\mathbf{d}=\mathbf{0}$. The remaining four imply that the $GL(4)$ matrix $A$ is the identity matrix. Similar reasoning applies in any dimension and also to the Euclidean case.

5.4 Reals and Vectors

Given a model $(\mathbb{P},B)\models_{2}\mathsf{BG}(4)$, we know, by theorem 62, that it is isomorphic to the standard coordinate structure $(\mathbb{R}^{4},B_{\mathbb{R}^{4}})$.

Using abstraction (or, equivalently, a quotient construction), we can extend $(\mathbb{P},B)$ with a new sort (or “universe” or carrier set) $\Re$ (of ratios) and operations $0,1,+,\times,\leq$ to a two-sorted structure $(\mathbb{P},\Re;B; 0,1,+,\times,\leq)$ where the reduct $(\Re;0,1,+,\times,\leq)$ is isomorphic to $\mathbb{R}$ (as an ordered field).⁴⁵ Call a triple $p,q,r$ of points a configuration just if $p\neq q$ and $p,q,r$ are collinear. This abstraction proceeds by the equivalence relation on configurations $(p,q,r)$ of proportionateness. In geometrical terms, there are three basic cases of proportionateness:⁴⁶

A real, or ratio, is then an equivalence class $[(p,q,r)]$ with respect to proportionateness, and $\Re$ is the set of these equivalence classes. One may define a zero $0$ as $[(p,q,p)]$ and a unit $1$ as $[(p,q,q)]$. One defines field operations $+$, $\times$, $\leq$ in terms of the corresponding operations on a fixed line (see Bennett 1995). One readily checks that the result is that $\Re$, with these operations, is a complete ordered field (and can then be identified with $\mathbb{R}$). Although we described this model theoretically, this construction can be “internalized” within $\mathsf{BG}(4)$ by adding suitable abstraction axioms (a “definition by abstraction”) for a new sort, with variables $\xi_{i}$ and a 3-place function symbol $\xi(p,q,r)$, and then explicitly defining $0$, $1$, $+$, $\times$, and $\leq$ on these new objects, and then proving that the resulting abstracta, i.e., the $\xi(p,q,r)$ for any configuration $p,q,r$, satisfy the second-order axioms for a complete ordered field.⁴⁷

We may further extend, with a new universe $\mathbb{V}$ (of displacements, or vectors) and operations $\mathbf{0},+,\cdot$, to a three-sorted structure $(\mathbb{P},\Re,\mathbb{V};B,0,1,+,\times,\leq;\mathbf{0},+,\cdot)$, where the reduct $(\mathbb{V},\Re;0,1,+,\times;\mathbf{0},+,\cdot)$ is isomorphic to $\mathbb{R}^{4}$ (as a vector space).⁴⁸ This abstraction proceeds by the equivalence relation on ordered pairs $(p,q)$ of equipollence: $(p,q)$ is equipollent to $(r,s)$ just if $p,q,s,r$ is a parallelogram:

A displacement, or vector, is then an equivalence class $[(p,q)]$ with respect to equipollence, and $\mathbb{V}$ is the set of these equivalence classes. An equivalence class $[(p,q)]$ is written $\mathbf{v}_{p,q}$. One may define the zero vector $\mathbf{0}$ as $\mathbf{v}_{p,p}$. One defines vector addition $+$ so that $\mathbf{v}_{p,q}+\mathbf{v}_{q,r}=\mathbf{v}_{p,r}$ holds (usually called Chasles’s Relation). One may define the scalar multiplication $\cdot$ so that, when $p\neq q$, $\alpha\cdot\mathbf{v}_{p,q}=\mathbf{v}_{p,r}$ just if $\alpha=[(p,q,r)]$; and, otherwise, $\alpha\cdot\mathbf{0}=\mathbf{0}$. One checks that the vector space axioms are true and that $\mathbb{V}$ is 4-dimensional.

Finally, by an explicit definition of an action $+:\mathbb{P}\times\mathbb{V}\to\mathbb{P}$, we can further extend to $(\mathbb{P},\Re,\mathbb{V};B;0,1,+,\times,\leq;\mathbf{0},+,\cdot;+)$ such that $(\mathbb{P},\mathbb{V},+)$ is isomorphic to the affine space $\mathbb{A}^{4}$.⁴⁹ The definition of the action $(p,\mathbf{v})\mapsto p+\mathbf{v}$ is: $q=p+\mathbf{v}$ iff $\mathbf{v}=\mathbf{v}_{p,q}$. One may then show that $+$ is a free and transitive action of $\mathbb{V}$ on $\mathbb{P}$. The affine space obtained in this way (basically, from the vector space $\mathbb{R}^{4}$, by “forgetting the origin”) is called $\mathbb{A}^{4}$.

The discussion and constructions above may be summarized in the following three theorems (I follow the usual practice of conflating the name of a structure with the name of its carrier set):

Theorem 67. $\Re$ is isomorphic to the complete ordered field $\mathbb{R}$.

Theorem 68. $\mathbb{V}$ is isomorphic to the vector space $\mathbb{R}^{4}$.

Theorem 69. $( \mathbb{P} , \mathbb{V} ,+)$ is isomorphic to the affine space $\mathbb{A}^{4}$.

The parameters “1” and “3” in $\mathsf{Gal}(1,3)$ mean: “1 time and 3 space dimensions.” Recall that an axiom system is called categorical when it has exactly one model up to isomorphism. Second-order Peano arithmetic, $\mathsf{PA}_2$, is categorical, its unique model being $(\mathbb{N},0,S,+,\times)$. The proof (essentially given in Dedekind 1888) is that if $M\models\mathsf{PA}_2$, we may define using Dedekind’s Recursion Theorem a function $\Phi:\mathbb{N}\to\text{dom}(M)$ by $\Phi(0)=0^M$ and, for all $n\in\mathbb{N}$, $\Phi(n+1)=S^M(\Phi(n))$. The axioms of $\mathsf{PA}_2$ then imply that $\Phi$ is a bijection, which is an isomorphism from $(\mathbb{N},0,S,+,\times)$ to $M$. In addition to $\mathsf{PA}_2$, the theory $\mathsf{ALG}$ of the complete ordered field is also categorical (essentially given in Huntington 1903; using methods developed in Dedekind 1872; Cantor 1897; Hölder 1901). Various second-order geometrical theories are also categorical. These include the systems denoted $\mathsf{BG}(4)$ and $\mathsf{EG}(3)$ below. Theorems 62 and 63 in appendix 5.2 establish the categoricity (and standard models) of these two systems. The proofs are due to Hilbert (1899), Veblen (1904), and Tarski (1959).↩︎
I think, informally, of a Galilean spacetime modally: a physically possible world with certain distinguished, or built-in, geometrical (spatio-temporal) relations. Such metaphysical issues, however, don’t matter here, as our whole discussion below is about models of $\mathsf{Gal}(1,3)$.↩︎
It is isomorphic to the standard four-dimensional affine space usually called $\mathbb{A}^{4}$ (see Gallier 2011), which is gotten from the vector space $\mathbb{R}^{4}$ by “forgetting its origin.” In Gallier’s notation, $\mathbb{A}^{4}$ is $(\mathbb{R}^{4},\mathbb{R}^{4},+)$, where the first $\mathbb{R}^{4}$ is the point set, the second $\mathbb{R}^{4}$ is the vector space, and $+$ is the action of vectors in $\mathbb{R}^{4}$ on points in $\mathbb{R}^{4}$. For the reader whose algebra is rusty, the notion of a group action is explained nicely in Dummit and Foote (2004, 41), Gallier (2011, 11), or Saunders (2013, 29).↩︎
A valuable semi-formal mathematical description of Galilean spacetime, incorporating what has just been said, is given in Arnold (1989, chap. 1).↩︎
Why do material points move (four-dimensionally) along these “grid lines” in Galilean spacetime? The physical answer is that such trajectories minimize the action. I.e., $\delta\int\ dt\ (\dot{q})^{2}=0$.↩︎
I have tried to write this paper so that it can be read by those unfamiliar with some of the somewhat arcane details of synthetic geometry. A very useful summary of the main ideas behind the construction of coordinate systems may be found in Burgess and Rosen (1997, 102–111). In my view, a very clear and nice introduction to the topic of affine and projective incidence geometry is Bennett (1995), where “geometric addition” and “multiplication” of points on a fixed line are explained clearly, and the core result is proved, that the line, with those operations, is a division ring (if Desargues’s Theorem is assumed) and a field (if Pappus’s Theorem is assumed). Notable reference works more generally are Coxeter (1969) and Hartshorne (2000). A fairly advanced treatment is Borsuk and Smielew (1960). Tarski’s papers (1959; and Tarski and Givant 1999) are very accessible. The first of these sketches the representation theorem for first-order Euclidean geometry and for the second-order Euclidean geometry $\mathsf{EG}(3)$ used below. Tarski focuses on the two-dimensional, first-order (“elementary”) case. The book Schwabhäuser, Szmielew and Tarski (1983) is very detailed (it is in German, and there is no English translation). Some recent works have implemented Tarski Euclidean geometry in theorem provers, just as one can implement arithmetic, set theory, and type theory in such provers. I have no doubt that this can, in principle, be generalized to our Galilean spacetime geometry and to one or other axiomatization of Minkowski spacetime geometry.↩︎
See Ketland (2022) for a formalization of the quicker proof in the Isabelle theorem prover.↩︎
In that book, Field’s approach is called “geometrical nominalism.” A technical difficulty that arises for Field’s program in Field (1980) concerning the problem of maintaining both a conservativeness condition and representation theorems is briefly described in remark 14 below.↩︎
As Einstein showed, the laws of gravitation amount to certain differential equations constraining $g_{ab}$ and the energy-momentum tensor $T_{ab}$. The “low energy limit” of Einstein’s field equation is Newton’s Law of Gravitation. Two standard textbooks on general relativity are Weinberg (1972) and Wald (1984).↩︎
Cf. Terence Tao (2008): “More generally, a coordinate system $\Phi$ can be viewed as an isomorphism $\Phi: A \to G$ between a given geometric (or combinatorial) object $\mathrm{A}$ in some class (e.g. a circle), and a standard object $\mathrm{G}$ in that class (e.g. the standard unit circle).”↩︎
To be clear, the synthetic and coordinate structures are isomorphic structures of the same signature, say $\sigma$. This is because it doesn’t make mathematical sense to talk of an isomorphism from $A$ to $B$ unless they are both $\sigma$-structures. E.g., it doesn’t make sense to say a group $(G,\oplus)$ is isomorphic to a ring $(R,+,\times)$ outside the special case where $\times$ is definable from $+$ or vice versa. Isomorphisms have to “match up” corresponding relations (operations and constants) in the signature. In logic, automated theorem proving, and so on, even seemingly small changes of the signature of the structures in question can make a large difference. For example, the structure $(\mathbb{N},0,S,+)$ is decidable (Presburger 1930), but $(\mathbb{N},0,S,+,\times)$ is undecidable (Gödel 1931; Tarski 1935). I’m grateful to a referee for mentioning this point, as related ones have arisen in the philosophy of physics.↩︎
See Trautman (1966), Stein (1967), Penrose (1968), Earman (1970, 1989), Ehlers (1973), Friedman (1983). One may also find mathematically precise descriptions in Arnold (1989, chap. 1) and in Kopczyński and Trautman (1992, 31–32).↩︎
Here, I am referring to such things as manifolds, diffeomorphisms, affine connections, tangent spaces, tensor fields, and whatnot. An excellent textbook on differential geometry, oriented towards advanced physics students, is Schutz (1980). Also, Malament (2012) and Wald (1984). For useful surveys of some of the surrounding philosophical issues, see Huggett and Hoefer (2015) (absolute vs. relational theories of spacetime) and DiSalle (2020) (inertial frames).↩︎
In Goldblatt (1987), a relation of “spacetime orthogonality,” $pq\perp rs$, is used, but $\perp$ and $\lambda$ are interdefinable, as Goldblatt shows.↩︎
Where the standard coordinate relation $\lambda_{\mathbb{R}^{4}}$ on $\mathbb{R}^{4}$ is defined as follows: for any $\mathbf{x},\mathbf{y}\in\mathbb{R}^{4}$, $\mathbf{x}\lambda_{\mathbb{R}^{4}}\mathbf{y}$ holds iff $\sum_{i=1}^{3}(x^i-y^i)^{2}-(x^{4}-y^{4})^{2}=0$ (i.e., the Minkowski interval is equal to $0$). I have set $c=1$.↩︎
In fact, to be a bit more accurate, I believe it is the “extended” Poincaré group, allowing global scaling, $x^{\mu} \mapsto \alpha x^{\mu}$ ($\alpha \neq 0$), of coordinates. This is because $( \mathbb{P} , \lambda)$ does not have a special “unit length.”↩︎
Field states his four axioms very briefly, in a footnote (1980, chaps. 6, 54, n.33). Field remarks, “Given the Szczerba-Tarski axiom on ‘Bet’, it is quite trivial to impose requirements on the two new primitives ‘Simul’ and ‘S-Cong’ so as to get the desired representation and uniqueness theorems” (1980, 54). Although Field takes a slightly different congruence relation as primitive (which he calls $\texttt{S-cong}$), I am reasonably sure that Field’s axiom system is definitionally equivalent to the one given here, $\mathsf{Gal}(1,3)$. I hope to publish the equivalence proof elsewhere. Burgess’s sketch of the geometry of Galilean spacetime (Burgess 1984; Burgess and Rosen 1997) uses our physical primitives and I believe Burgess must have separately established this equivalence.↩︎
$[\sim]^{4}$ is defined to be: $\{(p,q,r,s)\mid p\sim q\wedge p\sim r\wedge p\sim s\}$. See definition 12 below.↩︎
A 4-frame is an ordered quintuple of points that are not in the same 3-dimensional hypersurface. See definition 58 below.↩︎
We use the abbreviation $\mathbf{x}=(x^{1},\dots,x^{n})$ for $n$-tuples in $\mathbb{R}^{n}$. Similarly, for $\mathbf{y},\mathbf{z},\dots$. Hopefully, it will be clear that these don’t mean powers of $x$.↩︎
The second of these, in fact, generalizes to $n>1$ if we have a metric function: $B_{\mathbb{R}^{n}}(\mathbf{x},\mathbf{y},\mathbf{z}):=\Delta_{n}(\mathbf{x},\mathbf{y})+\Delta_{n}(\mathbf{y},\mathbf{z})=\Delta_{n}(\mathbf{x},\mathbf{z})$.↩︎
Regarding the definitions of $\mathbb{BG}^{n}$, $\mathbb{EG}^{n}$ and $\mathbb{G}^{(1,n)}$. These still make sense if we replace $\mathbb{R}$ in the definition by a Euclidean ordered field $F$ (an ordered field where all non-negative elements are squares). Cf. Szczerba and Tarski (1979, 160, Definition 1.5), who call a space $\mathbb{BG}^{n}(F)$ a “Cartesian affine space” over $F$.↩︎
Cf. the “interpretive principles” given in Malament (2012, 120–121).↩︎
Informally, we liberalize notation for point variables, occasionally using “$p$,” “$q$,” “$r$,” “$s$,” “$u$,” “$x$,” “$y$,” “$z$,” and the like, with natural number subscripts.↩︎
I use the moniker “$\mathsf{BG}$” to mean “betweenness geometry” ($n$ dimensions) for several reasons. First, because there doesn’t seem to be a standard name for these geometries. Second, they are sometimes called “affine geometries,” but the word “affine” has too many meanings, including two different meanings, each having nothing to do with the betweenness relation. These are “affine plane” (see, e.g., Bennett 1995) and “affine space” (see, e.g., Gallier 2011). Sometimes, the terminology “ordered geometry” is used (Pambuccian 2011). But “$\textsf{OG}$” seems to me ugly. Since the terminology is not entirely uniform, I use “betweenness geometry” and, hence, $\mathsf{BG}(4)$, etc. I should note that these axiom systems contain Euclid’s Parallel Postulate in some form.↩︎
The system $\mathsf{BG}(4)$ corresponds precisely to what Burgess called $\textsf{GEOM}_{4}$ in Burgess (1984). The system $\mathsf{BG}(4)$ also corresponds to what Szczerba and Tarski called $\textsf{GA}_{4}^{\ast}+\text{Euclid}$ in Szczerba and Tarski (1979, 1965). The term “$\textsf{GA}$” is used to mean a system of absolute or neutral geometry (i.e., without the Parallel Postulate), which is why (Euclid) is added. Note that (Euclid) is formulated entirely using $\texttt{Bet}$, and congruence does not appear. The subscript denotes the dimension, and the asterisk denotes that the axiom system is second-order; this means the Continuity Axiom is second-order rather than a scheme. A system essentially equivalent to $\textsf{GA}_{3}^{\ast}$ is studied carefully in the monograph Borsuk and Smielew (1960). The axioms of $\mathsf{BG}(4)$ are the result of simplifying the categorical system of “order axioms” given in Veblen (1904), where the relevant categoricity or representation theorem (i.e., our theorem 62 in appendix 5.2) was first given.↩︎
$\mathsf{EG}(3)$ itself corresponds to the second-order version of the three-dimensional version of Tarski’s system for synthetic Euclidean geometry in Tarski (1959), somewhat simplified in Tarski and Givant (1999). As with “$\mathsf{BG}$,” I use the moniker “$\mathsf{EG}$” to mean “Euclidean geometry.” In my notation, Tarski’s 1959 paper is mostly about the first-order theory $\mathsf{EG}_{0}(2)$, which is $\mathsf{EG}(2)$ “little’s brother.”↩︎
The relativization is more precisely defined as a translation $^{\circ}$, which acts as the identity on atomic formulas, which commutes with the Boolean logical connectives, and, for quantifiers, maps $\forall p\,\varphi$ to $(\forall p\sim z)\,\varphi^{\circ}$, maps $\exists p\,\varphi$ to $(\exists p\sim z)\,\varphi^{\circ}$, maps $\forall\mathsf{X}\,\varphi$ to $(\forall\mathsf{X}\subseteq\Sigma_{z})\,\varphi^{\circ}$, and maps $\exists\mathsf{X}\,\varphi$ to $(\exists\mathsf{X}\subseteq\Sigma_{z})\,\varphi^{\circ}$.↩︎
See also Borsuk and Smielew (1960, 7–8) on this topic.↩︎
A suitable “ambient set theory,” a system of axioms for the existence of sets, where the points will now be urelements or atoms (i.e., not sets or classes), and where comprehension, separation, and replacement schemes can be applied to any urelement predicate (e.g., $\texttt{Bet}$ and so on), is given in Ketland (2021). The ambient set theory is called $\textsf{ZFU}_{V(T)}$ in Field (1980, 17).↩︎
The axiom $\mathsf{Gal}5$ is so obvious that it occurred to me that it might indeed be provable from the remainder. However, I’ve not found a proof of this. So, I retain it. It is needed to show that the vector translation of a Galilean 4-frame is also a Galilean 4-frame (lemma 53 below).↩︎
The fact that if the points $p,q,r,s$ are simultaneous, then the points $p+\mathbf{v}$, $q+\mathbf{v}$, $r+\mathbf{v}$, and $s+\mathbf{v}$ are also simultaneous is given in lemma 45 below.↩︎
I follow Burgess (1984) in calling this theory $\mathsf{ALG}$. A standard axiomatization of $\mathsf{ALG}$ is given in Apostol (1967, 18, 20, 25). An equivalent axiomatization appears in Rudolf Carnap’s neglected textbook Carnap (1958, sec. 45, 183–185). See also Tarski (1946, 215) for a similar and equivalent formulation to (24), but Tarski uses the notion “the set $X$ strictly precedes the set $Y$” (using $<$ instead of $\leq$) and “$s$ separates the sets $X$ and $Y$” (again using $<$ instead of $\leq$). But these Continuity axioms are equivalent. And both are equivalent to the usual Dedekind cut axiom given in an analysis textbook: “any non-empty bounded subset of $\mathbb{R}$ has a supremum” (e.g., Apostol 1967, 25).↩︎
The precise definitions of the predicates $\texttt{co}_{n}$ are given in Szczerba and Tarski (1979, 190). (Szczerba and Tarski call these predicates $\mathbf{L}_{n}$.) The definition is recursive: for $n>1$, each $\texttt{co}_{n}$ is defined in terms of the previous ones. These definitions are due to Kordos (1969).↩︎
The required definitions of geometrical addition $+$ and geometrical multiplication $\times$ (which go back to Hilbert 1899) are given in Bennett (1995). The definition of the order on a fixed line in terms of $\texttt{Bet}$ is given in Tarski (1959, proof of theorem 1).↩︎
Perpendicularity $OX\perp^{\sim} OY$, for three distinct simultaneous points $O,X,Y$, is defined just as in definition 59 in appendix 5.2, but replacing the ordinary congruence predicate $\equiv$ by the sim-congruence predicate $\equiv^{\sim}$.↩︎
See also Malament (2009) for a nice exposition of these ideas.↩︎
Some geometry texts (e.g., Coxeter 1969, 213) will write $\overrightarrow{pq}$. E.g., Chasles’s Relation then becomes $\overrightarrow{pq}+\overrightarrow{qr}=\overrightarrow{pr}$.↩︎
I am grateful to a referee for bringing to my attention Saunders (2013), whose discussion of Galilean spacetime uses similar vector methods and the notion of affine space.↩︎
Where $\vec{\Phi}(p)$ is the triple $(\Phi^{1}(p),\Phi^{2}(p),\Phi^{3}(p))\in\mathbb{R}^{3}$, and $\Delta_{3}$ is the metric function on $\mathbb{R}^{3}$.↩︎
These axioms are given originally in Szczerba and Tarski (1965, 1979). See also Goldblatt (1987, 165) for the corresponding first-order theory, which we have called $\mathsf{BG}_{0}(4)$. Goldblatt calls this “the first-order theory of ordered affine fourfolds over real-closed fields.”↩︎
Burgess refers to such systems of points as “benchmarks”: Burgess and Rosen (1997, 107). For example, in the two-dimensional case, one may imagine marking three non-collinear points $O,X,Y$ on a bench. This will be a “2-frame” and will determine a two-dimensional coordinate system, with $O$ at the origin, $\ell(O,X)$ the “$x$-axis,” and $\ell(O,Y)$ the “$y$-axis.”↩︎
A similar definition, mutatis mutandis, can be applied to $\mathsf{BG}(n)$ in general and to $\mathsf{EG}(n)$ in general.↩︎
See also Borsuk and Smielew (1960) and Szczerba and Tarski (1965, 1979).↩︎
I included this appendix, in part, because I had difficulty locating the material elsewhere. One important textbook, Bennett (1995), where the definitions of addition $+$ and multiplication $\times$ on a line $\ell(p,q)$, and the proof that these induce a division ring (given Desargues’ Theorem) or a field (given Pappus’s Theorem), are explained very clearly, is out of print. Also, because I need, in the main part of the article, to refer to a couple of the summary theorems at the end of this appendix.↩︎
Burgess and Rosen (1997, 110) list two basic cases, our Case 1 and Case 3. In a sense, Case 3 is a limiting case of Case 2 by “sliding” the configuration $abc$ parallel to the three parallel lines until $a$ now coincides with $p$.↩︎
The details are given in Burgess (1984). What we’ve called “configurations,” Burgess calls “suitable configurations.” For the simple case of “extension by abstraction,” with a formula $\varphi(x,y)$ that can be shown to be an equivalence relation in the basic theory $T$, an extension of $T$ by abstraction is obtained by abstraction axioms (i): $\xi(x)=\xi(y)$ iff $\varphi(x,y)$; and (ii): $\forall\xi\,\exists x\,(\xi=\xi(x))$, where $\xi$ is a new variable sort, and $\xi(x)$ is a function symbol (which Burgess writes as “$[x]$”). See Burgess (1984, 381). Burgess shows (theorem 1.3) that this (indeed any) “extension by abstraction” is a conservative extension of the original theory $T$ and may be interpreted into the original theory. For the geometrical case, the abstraction axioms are (i): $\xi(p,q,r)=\xi(p',q',r')$ iff the configurations $p,q,r$ and $p',q',r'$ are proportionate; and (ii): $\forall\xi\,\exists p,q,r\,(p\neq q\wedge\texttt{co}_{1}(p,q,r)\wedge\xi=\xi(p,q,r))$. See Burgess (1984, 387, axioms (1) and (2)).↩︎
The two pluses ($+$) here have been overloaded: the first is the field addition, and the second is the vector addition.↩︎
The three pluses ($+$) here are overloaded: the first is the field addition; the second is the vector addition; and the third is the action, $(p,\mathbf{v})\mapsto p+\mathbf{v}$.↩︎

\(\mathsf{Gal}1\)	\(\mathsf{BG}(4)\).
\(\mathsf{Gal}2\)	\(\mathsf{EG}(3)^{\sim}\).
\(\mathsf{Gal}3\)	\(\sim\) is an equivalence relation.
\(\mathsf{Gal}4\)	\(\equiv^{\sim} \ \subseteq \ [\sim]^{4}\).
\(\mathsf{Gal}5\)	\(\equiv^{\sim}\) is translation-invariant.

\(\mathbb{BG}^{n}\)	Betweenness geometry in \(n\) dimensions over \(\mathbb{R}\)	\(:=(\mathbb{R}^{n},B_{\mathbb{R}^{n}}).\)
\(\mathbb{EG}^{n}\)	Euclidean space in \(n\) dimensions over \(\mathbb{R}\)	\(:=(\mathbb{R}^{n},B_{\mathbb{R}^{n}},\equiv_{\mathbb{R}^{n}}).\)
\(\mathbb{G}^{(1,n)}\)	Galilean spacetime in \(n+1\) dimensions over \(\mathbb{R}\)	\(:=(\mathbb{R}^{n+1},B_{\mathbb{R}^{n+1}},\sim_{\mathbb{R}^{n+1}},\equiv^{\sim}_{\mathbb{R}^{n+1}}).\)

\(\mathsf{Gal}1\)	\(\mathsf{BG}(4)\).
\(\mathsf{Gal}2\)	\(\mathsf{EG}(3)^{\sim}\).
\(\mathsf{Gal}3\)	\(\sim\) is an equivalence relation.
\(\mathsf{Gal}4\)	\(\equiv^{\sim} \ \subseteq \ [\sim]^{4}\).
\(\mathsf{Gal}5\)	\(\equiv^{\sim}\) is translation-invariant.

\(\equiv^{\sim}\)-Transitivity	\(\forall z[(\forall p,q,r,s,t,u\sim z)\,(pq\equiv^{\sim}rs\wedge pq\equiv^{\sim}tu\to rs\equiv^{\sim}tu)]\).
Pasch	\(\forall z[(\forall p,q,r,s,u\sim z)\,(\texttt{Bet}(p,q,r)\wedge\texttt{Bet}(s,u,q) \to(\exists x\sim z)\,(\texttt{Bet}(r,x,s)\wedge\texttt{Bet}(p,u,x)))]\).

\(B_{\mathbb{R}^{4}}\) represents \(B\) wrt \(\Phi\)	iff	for all \(p,q,r\in\mathbb{P}\): \(B(p,q,r) \leftrightarrow(\Phi(p),\Phi(q),\Phi(r))\in B_{\mathbb{R}^{4}}\).
\(\sim_{\mathbb{R}^{4}}\) represents \(\sim\) wrt \(\Phi\)	iff	for all \(p,q\in\mathbb{P}\): \(p\sim q \leftrightarrow\Phi(p)\sim_{\mathbb{R}^{4}}\Phi(q)\).
\(\equiv^{\sim}_{\mathbb{R}^{4}}\) represents \(\equiv^{\sim}\) wrt \(\Phi\)	iff	for all \(p,q,r,s\in\mathbb{P}\): \(pq\equiv^{\sim}rs \leftrightarrow\Phi(p)\Phi(q)\equiv^{\sim}_{\mathbb{R}^{4}} \Phi(r)\Phi(s)\).