2. Euclidean geometry

By three-dimensional elementary Euclidean geometry we mean a system of axioms that is collectively equivalent to the three-dimensional analog of Hilbert’s ([1902] Reference Hilbert1971) axioms for standard Euclidean plane geometry save his continuity axioms, the latter being the axioms that limit the models to isomorphic copies of two-dimensional Cartesian geometry over the ordered field $\mathbb{R}$ of real numbers. For our purpose, it is especially convenient to employ Tarski’s system ${{\cal P}_3}$ (Tarski Reference Tarski, Henkin, Suppes and Tarski1959, axioms ${A_1} - {A_{12}}$ and note 5; Schwabhäuser, Szmielew, and Tarski Reference Schwabhäuser, Szmielew and Tarski1983, 10–24), which is formulated as a first-order theory (with the equality symbol “=” treated as logical identity), in which points are the sole primitive individuals and the only nonlogical predicates employed in the axioms are a ternary primitive predicate B (where “ $Bxyz$ ” is read y lies between x and z, the case when y coincides with x or z not being excluded) and a quarternary primitive predicate $ \equiv $ (where “ $xy \equiv zu$ ” is read x is as distant from y as z is from u). The line through distinct points a and c in ${{\cal P}_3}$ is the set of all points b such that $Babc \vee Bbca \vee Bcab$ , and a segment $ac$ is the set of all points b such that $Babc$ , the segment being degenerate if $a = c$ .

The conception that bridges the gap between the domains of number and of Euclidean magnitude is the classical notion of a Cartesian space over an ordered field. In the latest version of Tarski’s framework (Schwabhäuser, Szmielew, and Tarski Reference Schwabhäuser, Szmielew and Tarski1983, 16), the three-dimensional version of this familiar concept assumes the following form.

A three-dimensional Cartesian space over an ordered field $\left\langle {F, + , \cdot , \lt ,0,1} \right\rangle $ is a structure ${C_3}\left( F \right) = :\left\langle {{A_F},{B_F},\!\!{ \equiv \!_F}} \right\rangle $ , where the betweenness relation ${B_F}$ and the equidistance relation $\!\!{ \equiv \!_F}$ are defined on ${A_F} = \left\{ {\left( {{x_1},{x_2},{x_3}} \right):{x_1},{x_2},{x_3} \in F} \right\}$ by the stipulations:

$${B_F}{\rm{ }}xyz\,{\rm{if}}\,{\rm{and}}\,{\rm{only}}\,{\rm{if}}\,{\rm{there}}\,{\rm{is}}\,{\rm{a}}\,\lambda \in F\,for\,which\,0 \le \lambda \le 1\,{\rm{and}}$$

${y_\alpha } - {x_\alpha } = \lambda \left( {{z_\alpha } - {x_\alpha }} \right)\;{\rm{f}}or\;\alpha = 1,2,3;$

$xy{ \equiv _F}uv\;{\rm{if\;and\;only\;if}}\;\sum\limits_{\alpha = 1}^3 {\left( {{x_\alpha } - {y_\alpha }} \right)^2} = \sum\limits_{\alpha = 1}^3 {\left( {{u_\alpha } - {v_\alpha }} \right)^2}.$

The relation between Cartesian spaces and models of ${{\cal P}_3}$ is given by the following well-known result that has its roots in the work of Hilbert (Reference Hilbert1899; [1902] Reference Hilbert1971).

Representation Theorem for ${{\cal P}_3}$ . M is a model of ${{\cal P}_3}$ if and only if M is isomorphic to ${C_3}\left( F \right)$ , where F is a Pythagorean ordered field—an ordered field F, in which $\sqrt {{a^2} + {b^2}} \in F$ whenever $a,b \in F$ .

The central ingredient in the proof of the only if portion of the above theorem is the result: if l is the line in a model M of ${{\cal P}_3}$ containing the distinct points o and e, then by letting the segments $oo$ and $oe$ serve as the zero (segment) and unit (segment), respectively, and by appealing to familiar geometric constructions, one can define operations ${ + _l}$ and ${ \cdot _l}$ on, and a relation ${ \lt _l}$ between, the directed segments of l (having o as an end point) so that the resulting structure

$F_{oe}^l = \left\langle {l,{ + _l},{ \cdot _l}{{, \lt }_l},oo,oe} \right\rangle ,$

is a Pythagorean ordered field that is within isomorphism independent of the choices of l, o, and $e \ne o$ . Employing $F_{oe}^l$ together with the triple of mutually perpendicular lines l, $l^\prime $ , and $l^{\prime \prime} $ of M having the point o in common, one then introduces a system of Cartesian coordinates and shows that M is isomorphic to ${C_3}(F_{oe}^l).$ Henceforth, we will refer to $F_{oe}^l$ (or an isomorphic copy thereof) as the characteristic ordered field of $M.$ ^{Footnote 2}

To obtain a categorical axiomatization of ordinary three-dimensional Euclidean geometry one need only supplement ${{\cal P}_3}$ with the Dedekind Continuity Axiom (Tarski Reference Tarski, Henkin, Suppes and Tarski1959, 18). Henceforth we will refer to the axiomatization thus obtained as ${{\cal E}_3}$ . Since the characteristic ordered field of a model of ${{\cal E}_3}$ is isomorphic to $\mathbb{R}$ , we now have the familiar

Representation Theorem for ${{\cal E}_3}$ . M is a model of ${{\cal E}_3}$ if and only if M is isomorphic to ${C_3}\left( \mathbb{R} \right)$ .

The constructions of our models of ${{\cal E}_3}$ make use of a familiar model-theoretic technique that constructs models vis-à -vis isomorphisms. Since the Dedekind Continuity Axiom is a second-ordered assertion, the discussion of these models takes place in classical second-order logic.

3. First model of ${{\cal E}_3}$ with extended points

For each point $x = \left( {{x_1},{x_2},{x_3}} \right) \in {A_\mathbb{R}} = {\mathbb{R}^3}$ , let ${\bf{x}}$ be a copy of $\left\langle {{A_\mathbb{R}},{B_\mathbb{R}},\!\!{ \equiv\! _\mathbb{R}}} \right\rangle = {C_3}\left( \mathbb{R} \right)$ indexed by x; that is, let

${\bf{x}}: = {C_3}{\left( \mathbb{R} \right)_x},$

which, on occasion, we will also write as $\left\langle {\mathbb{R}_x^3,{B_x},\!\!{ \equiv \!_x}} \right\rangle $ . To ensure that ${\bf{x}} \cap {\bf{y}} = \emptyset $ whenever $x \ne y$ , we may suppose, for example, that $\mathbb{R}_x^3: = \{ {a_x}: = (a,x):a \in {\mathbb{R}^3}\} $ . Further, let $\Pi = \left\{ {{\bf{x}}:x \in {\mathbb{R}^3}} \right\}$ and define betweenness and equidistance relations B and $ \equiv $ on $\Pi $ by stipulating: for all ${\bf{a}},{\bf{b}},{\bf{c}},{\bf{d}} \in \Pi $ ,

$B {\bf{abc}}\;{\rm{if\;and\;only\;if}}\;{B_\mathbb{R}}abc,$

${\bf{ab}} \equiv {\bf{cd}}\;{\rm{if\;and\;only\;if}}\;ab{ \equiv _\mathbb{R}}cd.$

Since the mapping $f:{\mathbb{R}^3} \to \Pi $ defined by the condition $f\left( x \right) = {\bf{x}} = {C_3}{\left( \mathbb{R} \right)_x}$ for all $x \in {\mathbb{R}^3}$ is an isomorphism from ${C_3}\left( \mathbb{R} \right)$ onto $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ , it follows from the above construction and the Representation Theorem for ${{\cal E}_3}$ that:

Moreover, since the characteristic ordered field of $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ is isomorphic to $\mathbb{R}$ , each segment of $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ is assigned a nonnegative real-valued length that is zero if and only if the segment is degenerate. Accordingly, we have

The reader will notice that if in the above construction and the arguments for Theorem 1 and Corollary 1, we replace ${\bf{x}}: = {C_3}{\left( \mathbb{R} \right)_x}$ with ${\bf{x}}: = \mathbb{R}_x^3$ , the resulting arguments continue to hold, and we obtain the following results:

The only difference between the two approaches is that in the first the points are models of ${{\cal E}_3}$ , and in the second they are the universes of the models. In our treatments of the subsequent two models of ${{\cal E}_3}$ , we will limit the discussion to the first approach, with the understanding that the second approach is available in those cases as well. In section 9, we will draw attention to a consequence that results from the difference in the two approaches.

4. Archimedean and non-Archimedean ordered fields and models of ${{\cal E}_3}$

Unlike the above construction, the next construction of a model of ${{\cal E}_3}$ containing extended points makes use of a non-Archimedean ordered field. To prepare the way for the latter construction as well as for some material in subsequent sections we will recall some of the basics of Archimedean and non-Archimedean ordered fields and their underlying ordered additive groups.

An ordered field $\left\langle {F, + , \cdot , \lt ,0,1} \right\rangle $ is said to be Archimedean if its ordered additive group $\left\langle {F, + , \lt ,0} \right\rangle $ satisfies the Archimedean condition: whenever a and b are nonzero members of F where $\left| a \right| \lt \left| b \right|$ , there is a positive integer n such that $n\left| a \right| \gt \left| b \right|$ . If F is not Archimedean, it is said to be non-Archimedean. Since every ordered field contains a unit element, written “ $1$ ,” an element a of F may be said to be infinitesimal if $\left| a \right| \lt 1/n$ for every positive integer n, and it may be said to be infinite if $\left| a \right| \gt n \cdot 1$ for every positive integer n. An ordered field is Archimedean if and only if it contains neither infinite nor nonzero infinitesimal elements. Non-Archimedean ordered fields, by contrast, contain infinite as well as nonzero infinitesimal elements. Zero is the sole infinitesimal element of an Archimedean ordered field.

If a and b are nonzero members of the ordered additive group of $\left\langle {F, + , \cdot , \lt ,0,1} \right\rangle $ , then a is said to be Archimedean equivalent to b (or finite relative to b) if there are positive integers m and n such that $m\left| b \right| \gt \left| a \right|$ and $n\left| a \right| \gt \left| b \right|$ . Archimedean equivalence partitions the elements of $F - \{ 0\} $ into disjoint classes called Archimedean classes. The term “Archimedean class” is intended to indicate that within a given class the Archimedean condition holds. If $a,b \in F$ are not Archimedean equivalent, then a is said to be infinitesimal (in absolute value) relative to b and b is said to be infinite (in absolute value) relative to a, if $\left| a \right| \lt \left| b \right|$ . In accordance with these conventions, $0$ is infinitesimal (in absolute value) relative to every member of $F - \{ 0\} $ . Given the absence of a unit element, the notions of infinite and nonzero infinitesimal members of $\left\langle {F, + , \lt ,0} \right\rangle $ are not well-defined. On the other hand, the infinite and infinitesimal members of $\left\langle {F, + , \cdot , \lt ,0,1} \right\rangle $ are the members of the field that are (in absolute value) infinite and infinitesimal, respectively, relative to the unit element.

A model of ${{\cal P}_3}$ may be said to be Archimedean if and only if its characteristic ordered field is Archimedean (see section 2). Therefore, since an ordered field is Archimedean if and only if it is isomorphic to a subfield of $\mathbb{R}$ , every model of ${{\cal E}_3}$ is Archimedean. As such, the nondegenerate segments of a model of ${{\cal E}_3}$ are all finite relative to one another and infinite relative to the degenerate segments of the model.

5. The non-Archimedean ordered field ${\cal L}$

The construction that underlies our second model can be carried out using any non-Archimedean Pythagorean ordered field that contains an isomorphic copy of the ordered field of real numbers. The following familiar example of such a field was introduced by Tullio Levi-Civita in the first of his groundbreaking works (Reference Levi-Civita1892–93; Reference Levi-Civita1898) that placed Giuseppe Veronese’s pioneering investigation of non-Archimedean geometry (Reference Veronese1891) on an algebraic foundation.

Let ${\cal L}$ be the collection of all power series

$\sum\limits_{\alpha \lt \beta } {r_\alpha }{t^{{\gamma _\alpha }}}$

where $\left\{ {{\gamma_\alpha }:\alpha \lt \beta \le\omega} \right\}$ is a (possibly empty) strictly decreasing sequence of members of $\mathbb{R}$ that is coinitial with $\mathbb{R}$ if $\beta = \omega $ , and $\left\{ {{r_\alpha }:\alpha \lt \beta } \right\}$ is a sequence of members of $\mathbb{R} - \left\{ 0 \right\}$ . ${\cal L}$ is a non-Archimedean ordered field (with the empty series serving as the zero of the field) when the order is defined lexicographically and the sums and products are defined à la polynomials, with ${t^\gamma } \cdot {t^{\gamma \prime }} = {t^{\gamma + \gamma \prime }}$ for all $\gamma ,\gamma \prime \in \mathbb{R}$ . If we insert “dummy” terms with zeros for coefficients to permit a uniform representation of members of ${\cal L}$ , then these conditions (written ${ + _{\cal L}},{ \cdot _{\cal L}}$ , and ${ \lt _{\cal L}}$ ) may be stated more formally as follows, where $ + $ , $ \cdot $ , and $ \lt $ designate the standard addition, multiplication, and order in $\mathbb{R}$ .

$\sum\limits_{\gamma \in \mathbb{R}} {a_\gamma }{t^\gamma }{ + _{\cal L}}\sum\limits_{\gamma \in \mathbb{R}} {b_\gamma }{t^\gamma } = \sum\limits_{\gamma \in \mathbb{R}} \left( {{a_\gamma } + {b_\gamma }} \right){t^\gamma },$

$\sum\limits_{\gamma \in \mathbb{R}} {a_y}{t^\gamma }{ \cdot _{\cal L}}\sum\limits_{\gamma \in \mathbb{R}} {b_y}{t^\gamma } = \sum\limits_{\gamma \in \mathbb{R}} \left[ {\sum\limits_{{\scriptstyle(\mu ,\nu ) \in \mathbb{R} \times \mathbb{R}\atop\scriptstyle\mu + \nu = \gamma }} {a_\mu }{b_\nu }} \right]{t^\gamma },$

$\sum\limits_{\gamma \in \mathbb{R}} {a_\gamma }{t^\gamma }{ \lt _{\cal L}}\sum\limits_{\gamma \in \mathbb{R}} {b_\gamma }{t^\gamma },\;{\rm{if}}\;{a_\gamma } = {b_\gamma }\;{\rm{for\;all}}\;\gamma \gt {\rm{some}}\;\tau \in \mathbb{R}\;{\rm{and}}\;{a_\tau } \lt {b_\tau }.$

In virtue of the lexicographical ordering, the infinitesimals of ${\cal L}$ are the members of ${\cal L}$ whose greatest exponent ${\gamma _0}$ is negative. Since 0 corresponds to the empty series, this vacuously holds for 0.

6. Second model of ${{\cal E}_3}$ with extended points

Let ${\cal L}$ be defined as above, and, henceforth, by $\hat {\mathbb{R}}$ we mean the isomorphic copy of $\mathbb{R}$ in ${\cal L}$ consisting of $\left\{ 0 \right\} \cup \left\{ {x \in {\cal L}:x = r{t^0}{\kern 1pt} {\rm{for}}{\kern 1pt} {\rm{some}}{\kern 1pt} r \in \mathbb{R} - \left\{ 0 \right\}} \right\}$ with the addition, multiplication, and order inherited from ${\cal L}$ . First note that, since ${\cal L}$ is a Pythagorean ordered field (e.g., Robinson Reference Robinson1973, 92), ${C_3}\left( {\cal L} \right)$ is a model of ${{\cal P}_3}$ . Moreover, since ${\hat {\mathbb{R}}} \subset {\cal L}$ and ${\hat {\mathbb{R}}}$ is isomorphic to $\mathbb{R}$ , ${C_3}( {{\hat {\mathbb{R}}}} ) \subset {C_3}( {\cal L} )$ and ${C_3}( {{\hat {\mathbb{R}}}} )$ is isomorphic to ${C_3}\left( \mathbb{R} \right).$

Let ${{\cal L}_I}$ be the set of infinitesimals of ${\cal L}$ , and for each point $x = \left( {{x_1},{x_2},{x_3}} \right) \in {{\hat {\mathbb{R}}}^3}$ , let

${\bf{x}}: = \left\{ {\left( {{x_1} + {\delta _1},{x_2} + {\delta _2},{x_3} + {\delta _3}} \right) \in {{\cal L}^3}:{\delta _1},{\delta _2},{\delta _3} \in {{\cal L}_I}} \right\},$

together with the restrictions thereto of the betweenness and equidistance relations ${B_{\cal L}}$ and $\!\!{ \equiv _{\cal L}}$ from ${C_3}\left( {\cal L} \right)$ . Thus, for each point $x \in {{\hat {\mathbb{R}}}^3}$ , the universe of ${\bf{x}}$ is the infinite set of all points in ${C_3}\left( {\cal L} \right)$ that are an infinitesimal distance from x, including x itself. Using this, one may readily show: ${\bf{x}} \cap {\bf{y}} = \emptyset $ for distinct $x,y \in {{\hat {\mathbb{R}}}^3}$ ; and for each $x \in {{\hat {\mathbb{R}}}^3}$ , ${\bf{x}}$ is a convex region of ${C_3}\left( {\cal L} \right)$ in the sense that for all $a,b,c \in {C_3}\left( {\cal L} \right)$ , $b \in {\bf{x}}$ whenever $a,c \in {\bf{x}}$ and ${B_{\cal L}}abc$ . Moreover, ${\bf{x}}$ is three-dimensional, as is evident from its containment of the three mutually orthogonal (nondegenerate) segments of ${C_3}\left( {\cal L} \right)$ having endpoints

$({x_1},{x_2},{x_3}),({x_1} + \delta ,{x_2},{x_3})$

$({x_1},{x_2},{x_3}),({x_1},{x_2} + \delta ,{x_3})$

$({x_1},{x_2},{x_3}),({x_1},{x_2},{x_3} + \delta ),$

where $\delta $ is a nonzero member of ${{\cal L}_I}$ .

Now let

${\hat \Pi}:= \left\{ {{\bf{x}}:x \in {{{\hat {\mathbb{R}}}}^3}} \right\}$

and define betweenness and equidistance relations $\hat B$ and $\hat \equiv $ on $\hat \Pi $ by the conditions: for all ${\bf{a}},{\bf{b}},{\bf{c}},{\bf{d}} \in \hat \Pi $ ,

$\hat B{\bf{abc}}\;{\rm{if\;and\;only\;if}}\;{B_{\cal L}}abc,$

${\bf{ab}}\hat \equiv {\bf{cd}}\;{\rm{if\;and\;only\;if}}\;ab{ \equiv _{\cal L}}cd.$

Also let $f:{{\hat {\mathbb{R}}}^3} \to \hat \Pi $ be the mapping defined by the condition: $f\left( x \right) = {\bf{x}}$ for all $x \in {{\hat {\mathbb{R}}}^3}$ . Plainly, f is an isomorphism from ${C_3}\left( {{\hat {\mathbb{R}}}} \right)$ onto $\left\langle {\hat \Pi ,\hat B,\!\!\hat \equiv } \right\rangle $ ; and so, since ${C_3}\left( {{\hat {\mathbb{R}}}} \right)$ is isomorphic to ${C_3}\left( \mathbb{R} \right)$ , it follows that $\left\langle {\hat \Pi ,\hat B,\hat \equiv } \right\rangle $ is isomorphic to ${C_3}\left( \mathbb{R} \right)$ . But then, in virtue of the Representation Theorem for ${{\cal E}_3}$ and the just described construction, we have

Moreover, by invoking the argument employed in section 3 to establish Corollary 1, we have

7. Hjelmslev’s nilpotent infinitesimalist continuum

As in the second construction of a model of ${{\cal E}_3}$ with extended points, in our third construction the points are convex three-dimensional regions of an overarching geometric space. Here the overarching space is an affine Hjelmslev space over the lexicographically (totally) ordered ring $\mathbb{R}[\varepsilon ]$ of dual numbers, the three-dimensional counterpart of the nilpotent infinitesimalist geometric continuum introduced by Johannes Hjelmslev (Reference Hjelmslev1923). To prepare the way for the construction, we begin with a brief overview of the system of dual numbers and the three-dimensonal affine Hjelmslev space defined on it.^{Footnote 3}

The system $\mathbb{R}[\varepsilon ]$ of dual numbers, which was introduced by William Clifford (Reference Clifford1873), is a commutative ring with identity consisting of

$\{ {r_0} + {r_1}\varepsilon :{r_0},{r_1} \in \mathbb{R}\} $

with sums and products defined in the manner of polynomials, with $\varepsilon \ne 0$ and ${\varepsilon ^2} = 0$ . Thus, for any two dual numbers ${a_0} + {a_1}\varepsilon $ and ${b_0} + {b_1}\varepsilon $

$({a_0} + {a_1}\varepsilon ) + ({b_0} + {b_1}\varepsilon ) = ({a_0} + {b_0}) + ({a_1} + {b_1})\varepsilon $

$({a_0} + {a_1}\varepsilon )({b_0} + {b_1}\varepsilon ) = {a_0}{b_0} + ({a_0}{b_1} + {a_1}{b_0})\varepsilon .$

The element $\varepsilon $ in $\mathbb{R}[\varepsilon ]$ is a nilpotent, an element x such that ${x^n} = 0$ for some positive integer n. The least n for which ${x^n} = 0$ is the index of nilpotency of x. The index of nilpotency of $\varepsilon $ in $\mathbb{R}[\varepsilon ]$ is 2. Nonzero nilpotent elements of index n have many applications in algebra, one of them being a convenient way of representing quantities up to infinitesimal order n. When nilpotents are thus interpreted, they are referred to as nilpotent infinitesimals. The number systems we have hitherto considered have no nonzero nilpotent elements.

Inspired by the view of the pre-Socratic philosopher Protagoras, as recounted in Aristotle’s Metaphysics (III: 2, 998a), Hjelmslev held that the axiom that two straight lines always share at most one point is incompatible with perceptual experience, as is the assertion that a circle and a line tangent to it meet at a single point (Hjelmslev Reference Hjelmslev1923, 1–2). This led him to devise a “geometry of reality” (Hjelmslev Reference Hjelmslev1916) or a “natural geometry” (Hjelmslev Reference Hjelmslev1923), as he later called it, whose subject is the lines and circles of perception constructed with real rulers and compasses. He further devised (Hjelmslev Reference Hjelmslev1923, 12–13) the above-mentioned abstraction of the latter, coordinated by $\mathbb{R}[\varepsilon ]$ , that is a prototype of what are today called affine Hjelmslev geometries. In these geometries, which are coordinated by affine Hjelmslev rings, while each pair of distinct points has a line joining them, it need not be unique. Indeed, a pair of distinct points may lie on a pair of distinct lines; when this happens the points are said to be neighboring points, and the lines are said to be neighboring lines, the two notions of neighbor being equivalence relations. Remote (or nonneighboring) points, by contrast, are joined by a unique line, and remote (or nonneighboring) lines that intersect, intersect in a unique point. In these geometries a circle and a line tangent to it intersect in a nondegenerate “infinitesimal segment” of neighboring points.

Formally speaking, in their standard formulations, affine Hjelmslev planes are structures $\left\langle {\mathbb{P},\mathbb{L},\parallel ,I} \right\rangle $ (satisfying a set of axioms—see references below) where $\mathbb{P}$ is a set of points, $\mathbb{L}$ is a set of subsets of $\mathbb{P}$ called lines, $\parallel $ is a parallel relation defined on $\mathbb{L}$ , and I is an incidence relation defined on pairs $(a,b) \in \mathbb{P} \times \mathbb{L}$ . Moreover, and of central importance, the axioms ensure that the geometrical relational structure $\left\langle {{\mathbb{P}^*},{\mathbb{L}^*},{\parallel ^*},{I^*}} \right\rangle $ , with the induced relations ${\parallel ^*}$ and ${I^*}$ , obtained from $\left\langle {\mathbb{P},\mathbb{L},\parallel ,I} \right\rangle $ by replacing each $p \in \mathbb{P}$ and each $l \in \mathbb{L}$ with the equivalence classes consisting of all neighboring points of p and all neighboring lines of l, respectively, is isomorphic to an ordinary affine geometric counterpart, in which two points determine a line and lines that intersect do so in a unique point. In the case of Hjelmslev’s aforementioned prototype, which is coordinated by $\mathbb{R}[\varepsilon ]$ , the affine counterpart is (as Hjelmslev observed) isomorphic to the standard affine Euclidean plane over $\mathbb{R}$ . It is the existence of this isomorphism that motivates our third construction.

Hjelmslev’s plane over $\mathbb{R}[\varepsilon ]$ , henceforth ${H_2}\left( {\mathbb{R}[\varepsilon ]} \right)$ , consists of all ordered pairs $(A,B) \in \mathbb{R}[\varepsilon ] \times \mathbb{R}[\varepsilon ]$ . As usual, a straight line of ${H_2}\left( {\mathbb{R}[\varepsilon ]} \right)$ is defined by a first-degree equation

$Ax + By + C = 0,$

where now, however, $A,B,C \in \mathbb{R}[\varepsilon ]$ , it being understood that at least one of $A,B$ is not in the ideal $\mathbb{R}{[\varepsilon ]_I} = \{ 0 + r\varepsilon :r \in \mathbb{R}\} $ of nilpotent infinitesimal elements of $\mathbb{R}[\varepsilon ]$ . In ${H_2}\left( {\mathbb{R}[\varepsilon ]} \right)$ , a pair of points $({a_1},{a_2})$ and $({b_1},{b_2})$ are neighbors if and only if ${b_1} - {a_1}$ and ${b_2} - {a_2}$ are elements of $\mathbb{R}{[\varepsilon ]_I}$ .

$\mathbb{R}[\varepsilon ]$ admits a relational expansion to a non-Archimedean totally ordered ring, where the order $ \lt $ is defined lexicographically by the condition: ${a_0} + {a_1}\varepsilon \lt {b_0} + {b_1}\varepsilon $ if ${a_0}{ \lt _\mathbb{R}}{b_0}$ or ${a_0} = {b_0}$ and ${a_1}{ \lt _\mathbb{R}}{b_1}$ . In virtue of the just noted ordering, the points on a line of ${H_2}\left( {\mathbb{R}[\varepsilon ]} \right)$ are themselves totally ordered and admit a natural betweenness relation ${B_{\mathbb{R}[\varepsilon ]}}$ defined by the condition: for all points $x,y,z$ on a line l of ${H_2}\left( {\mathbb{R}[\varepsilon ]} \right)$ totally ordered by ${ \lt _l}$ , y lies between x and z if either $x{ \lt _l}y{ \lt _l}z$ or $z{ \lt _l}y{ \lt _l}x$ .^{Footnote 4}

Examples of neighboring lines in ${H_2}\left( {\mathbb{R}[\varepsilon ]} \right)$ are

$y = 0\;{\rm{and}}\;y = \varepsilon x,$

which intersect at all neighboring points $(r\varepsilon ,0)$ where $r \in \mathbb{R}$ . And similarly, the circle defined by the equation

$A{x^2} + B{y^2} = 1$

intersects the line defined by the equation

$y = 1$

at all neighboring points $(r\varepsilon ,1)$ where $r \in \mathbb{R}$ . In each of these cases, the set of neighboring points constituting the segment of intersection is isomorphic to $\mathbb{R}$ considered as an ordered set (Hjelmslev Reference Hjelmslev1923, 13).

Inspired by these and related ideas of Hjelmslev (Reference Hjelmslev1923, Reference Hjelmslev1929), the axiomatic theory of Hjelmslev planes (and corresponding theory of Hjelmslev rings) was developed by Klingenberg (Reference Klingenberg1954, Reference Klingenberg1954a, Reference Klingenberg1955) and was extended to higher dimensional spaces by Kreuzer (Reference Kreuzer1987, Reference Kreuzer1988). Our third construction, to which we now turn, makes use of the three-dimensional analog of ${H_2}\left( {\mathbb{R}[\varepsilon ]} \right)$ , henceforth denoted ${H_3}\left( {\mathbb{R}[\varepsilon ]} \right)$ , where the basic ideas and concepts of ${H_2}\left( {\mathbb{R}[\varepsilon ]} \right)$ readily extend to the three-dimensional case. In particular, in ${H_3}\left( {\mathbb{R}[\varepsilon ]} \right)$ , a pair of points $({a_1},{a_2},{a_3})$ and $({b_1},{b_2},{b_3})$ are neighbors if and only if ${b_1} - {a_1}$ , ${b_2} - {a_2}$ , and ${b_3} - {a_3}$ are elements of $\mathbb{R}{[\varepsilon ]_I}$ , and given a point and a line in a plane of ${H_3}\left( {\mathbb{R}[\varepsilon ]} \right)$ there is one line perpendicular to the line through the point (Kreuzer Reference Kreuzer1987, Reference Kreuzer1988).

8. Third model of ${{\cal E}_3}$ with extended points

Let $\tilde {\mathbb{R}}$ be the isomorphic copy of $\mathbb{R}$ in $\mathbb{R}[\varepsilon ]$ consisting of $\{ r + 0\varepsilon :r \in \mathbb{R}\} $ with the addition, multiplication, and order inherited from $\mathbb{R}[\varepsilon ]$ , and for each $x = \left( {{x_1},{x_2},{x_3}} \right) \in {{\tilde {\mathbb{R}}}^3}$ , let

${\bf{x}}: = \left\{ {\left( {{x_1} + {\delta _1},{x_2} + {\delta _2},{x_3} + {\delta _3}} \right) \in \mathbb{R}{{[\varepsilon ]}^3}:{\delta _1},{\delta _2},{\delta _3} \in \mathbb{R}{{[\varepsilon ]}_I}} \right\}$

together with the restriction thereto of the aforementioned betweennesss relation ${B_{\mathbb{R}[\varepsilon ]}}$ defined on ${H_3}\left( {\mathbb{R}[\varepsilon ]} \right)$ . Finally, let

${\tilde \Pi}: = \left\{ {{\bf{x}}:x \in {{{\tilde {\mathbb{R}}}}^3}} \right\}.$

For each $x \in {{\tilde {\mathbb{R}}}^3}$ , ${\bf{x}}$ is the (infinite) set of all points in ${H_3}\left( {\mathbb{R}[\varepsilon ]} \right)$ that are neighbors of x, including x itself. Since the neighbor relation on points is an equivalence relation that partitions $\mathbb{R}{[\varepsilon ]^3}$ into disjoint classes, it follows that for distinct $x,y \in {{\tilde {\mathbb{R}}}^3}$ , ${\bf{x}} \cap {\bf{y}} = \emptyset $ and

$\bigcup\limits_{{\bf{x}} \in \tilde \Pi } {\bf{x}} = \mathbb{R}{[\varepsilon ]^3}.$

Moreover, by a result of Baker, Lane, Lorimer and Laxton (Reference Baker, Lane, Lorimer and Laxton1984, 20–22)^{Footnote 5}, for each $x \in {{\tilde {\mathbb{R}}}^3}$ , ${\bf{x}}$ is a convex subclass of ${H_3}\left( {\mathbb{R}[\varepsilon ]} \right)$ in the sense defined in section 6. Furthermore, ${\bf{x}}$ is three-dimensional, in virtue of its containment of triples of mutually orthogonal nondegenerate segments of ${H_3}\left( {\mathbb{R}[\varepsilon ]} \right)$ , such as:

$\left\{ {\left( {{x_1} + \delta ,{x_2},{x_3}} \right) \in \mathbb{R}{{[\varepsilon ]}^3}:0 \le \delta \le \varepsilon } \right\}$

$\left\{ {\left( {{x_1},{x_2} + \delta ,{x_3}} \right) \in \mathbb{R}{{[\varepsilon ]}^3}:0 \le \delta \le \varepsilon } \right\}$

$\left\{ {\left( {{x_1},{x_2},{x_3} + \delta } \right) \in \mathbb{R}{{[\varepsilon ]}^3}:0 \le \delta \le \varepsilon } \right\}.$

By now mimicking the constructions employed in sections 3 and 6, we define betweenness and equidistance relations $\tilde B$ and $\tilde \equiv $ on $\tilde \Pi $ by the conditions: for all ${\bf{a}},{\bf{b}},{\bf{c}},{\bf{d}} \in \tilde \Pi $ ,

$\tilde B{\bf{abc}}\;{\rm{if\;and\;only\;if}}\;{B_\mathbb{R}}abc,$

${\bf{ab}}\tilde \equiv {\bf{cd}}\;{\rm{if\;and\;only\;if}}\;ab{ \equiv _\mathbb{R}}cd;$

and by essentially repeating the arguments employed in the just noted sections we obtain the following analogs for $\left\langle {\tilde \Pi ,\tilde B,\tilde \equiv } \right\rangle $ of the results for $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ and $\left\langle {\hat \Pi ,\hat B,\hat \equiv } \right\rangle $ .

9. Mathematico-philosophical reflections

Whereas the notions of length, area, and volume measure were introduced to quantify our preanalytic notions of one-dimensional, two-dimensional, and three-dimensional spatial extension, the relation between the standard geometrical notions and the preanalytic, metageometric/metaphysical notions are not quite what is often assumed. Indeed, what our models illustrate is that it is merely the infinitesimality of degenerate segments relative to their nondegenerate counterparts (see section 4), rather than the absence of extension of points, that is implied both by the axioms of ordinary Euclidean geometry and these segments’ null lengths.^{Footnote 6} Thus, from a contemporary standpoint, contrary to Euclid’s intuition, a point of Euclidean geometry may very well have parts, indeed even proper parts. Moreover, as the above remarks suggest, the number zero functions quite differently as a cardinal number than as a measure number in the system of reals. So, for example, whereas a set containing zero members has no member at all, an event having probability zero—such as a randomly thrown point-tipped dart landing on a specified point of the unit interval of $\mathbb{R}$ —may very well transpire, and perhaps more surprisingly still, a degenerate segment having length zero may contain a point having an internal structure that models all of Euclidean space.^{Footnote 7}

On the other hand, as we mentioned above, whereas the axioms of ${{\cal E}_3}$ make assertions about points, they are entirely reticent about what internal structure, if any, the points themselves have. Indeed, this is simply not the province of Euclidean geometry, at least not in its contemporary guise. Any additional assumptions about the nature of the points other than those derivable from the axioms for ${{\cal E}_3}$ are by their very nature extra-Euclidean. However, as we also previously mentioned, this is not to suggest that it would be impossible to suitably expand models of ${{\cal E}_3}$ that would have provable consequences about any internal structure the points might have. In the case of the models of ${{\cal E}_3}$ discussed above this can be carried out in a variety of fashions, including the following two in which the models are expanded to suitable many-sorted structures with supplemental universes and relations defined thereon. Since the constructions are similar in all three cases, we will simply illustrate them with $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ .

In the first approach, which is also the approach that more literally captures the ideas expressed in Theorem 1 and Theorem ${1^\dagger }$ , one expands the Euclidean structure $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ to the extra-Euclidean structure

$\left\langle {\Pi ,B,\!\!\equiv \!,(\left\langle {\mathbb{R}_x^3,{B_x},\!\!{ \equiv\! _x}} \right\rangle {)_{x \in {\mathbb{R}^3}}}} \right\rangle ,$

and in the second approach, one expands $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ to the extra-Euclidean structure

$\left\langle {\Pi ,B,\!\!\equiv ,{\Pi ^*},{B^*},{ \equiv ^*}} \right\rangle ,$

where

${\Pi ^*} = \bigcup\limits_{x \in {\mathbb{R}^3}} \mathbb{R}_x^3,\,\,{B^*} = \bigcup\limits_{x \in {\mathbb{R}^3}} {B_x}\,\,,{\rm{and}}\,{ \equiv ^*}\,\, = \bigcup\limits_{x \in {\mathbb{R}^3}} \!{ \equiv \!_x}.$

Thus, whereas in the first approach one adds for each point ${\bf{x}} \in \Pi $ a distinct universe $\mathbb{R}_x^3$ and distinct betweenness and equidistance relations ${B_x}$ and ${ \equiv\!_x}$ defined thereon, in the second approach one adds a single composite universe together with the corresponding composite betweenness and equidistance relations defined on it. While the structures emerging from the two approaches are formally distinct, the difference is largely a difference in methods of bookkeeping, since each individually adjoined structure in the first approach constitutes a recognizable convex substructure of $\left\langle {{\Pi ^*},{B^*},\!\!{ \equiv ^*}} \right\rangle $ in the second. Moreover, whether one interprets $\Pi $ as a set of models of ${{\cal E}_3}$ , as in Theorem 1, or as the set of the universes of the just noted models, as in Theorem ${1^\dagger }$ , the above two expansions of $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ are alternative ways of formalizing the extra-Euclidean structures implicit in each of those theorems.

If we adopt the first and more literal approach, then Theorem 1 and Theorem ${1^\dagger }$ assume the following explicit forms:

The above-described many-sorted expansions of $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ shed light on the internal structure of the points in $\Pi $ while leaving the model and, in particular, its points in place. Loosely speaking, whereas $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ provides the macroperspective of the geometric framework, $\left\langle {\Pi ,B,\!\!\equiv ,(\left\langle {\mathbb{R}_x^3,{B_x},\!\!{ \equiv _x}} \right\rangle {)_{x \in {\mathbb{R}^3}}}} \right\rangle $ and $\left\langle {\Pi ,B,\!\!\equiv ,{\Pi ^*},{B^*},\!\!{ \equiv ^*}} \right\rangle $ provide macro/microperspectives. There is also a complementary one-sorted structure that sheds further light on the microperspective without expanding the model itself. Since the basic construction is applicable to all three of our examples, again we will only illustrate it for the case of $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ .

Instead of expanding the structure $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ with the universe ${\Pi ^*}$ and the relations ${B^*}$ and $\!\!{ \equiv ^*}$ defined thereon, one begins with the structure $\left\langle {{\Pi ^*},{B^*},\!\!{ \equiv ^*}} \right\rangle $ and expands it to the structure $\left\langle {{\Pi ^*},B^\prime , \equiv ^\prime ,{B^*},\!\!{ \equiv ^*}} \right\rangle $ , where $B^\prime $ and $ \equiv ^\prime $ are defined by the stipulations:

$B^\prime {a_x}{b_y}{c_z}\;{\rm{if\;and\;only\;if}}\;{B_\mathbb{R}}xyz,$

${a_w}{b_x} \equiv ^\prime {c_y}{d_z}\;{\rm{if\;and\;only\;if}}\;wx\!\!{ \equiv _\mathbb{R}}yz,$

where $w,x,y,z$ range over ${\mathbb{R}^3}$ and ${a_w},{b_x},{c_y}$ , and ${d_z}$ range over $\mathbb{R}_w^3,\mathbb{R}_x^3,\mathbb{R}_y^3$ , and $\mathbb{R}_z^3$ , respectively. Unlike the tuples that comprise ${B^*}$ and $\!\!{ \equiv ^*}$ , those that comprise $B^\prime $ and $ \equiv ^\prime $ do not solely relate members of individual $\mathbb{R}_x^3$ ’s. Moreover, as is evident from the construction, there is an epimorphism (i.e., a surjective homomorphism) f from $\left\langle {{\Pi ^*},B^\prime , \equiv ^\prime } \right\rangle $ onto $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ , and, hence, $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ is isomorphic to the quotient structure $\left\langle {{\Pi ^*}/R,{{B^\prime }_R},{\!\!{ \equiv ^\prime }_R}} \right\rangle $ defined by the congruence relation R generated by f; that is, the structure $\left\langle {{\Pi ^*}/R,{{B^\prime }_R},{\!\!{ \equiv ^\prime }_R}} \right\rangle $ where ${\Pi ^*}/R$ is the set of all equivalence classes $[x] = \{ y \in {\Pi ^*}:f(y) = f(x)\} $ , ${B^\prime _R}$ is the set of all triples $([x],[y],[z])$ for which $(x,y,z) \in B^\prime $ , and $\!\!{ \equiv ^\prime _R}$ is the set of all quadruples $([w],[x],[y],[z])$ for which $(w,x,y,z) \in \equiv ^\prime $ (e.g., Monk Reference Monk1976, 385–86). Since f is not 1–1, unlike $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ , $\left\langle {{\Pi ^*},B^\prime , \equiv ^\prime } \right\rangle $ is not a model of ${{\cal E}_3}$ . On the other hand, $\left\langle {{\Pi ^*},B^\prime , \equiv ^\prime } \right\rangle $ may be regarded as a pre-Euclidean geometry in the sense that the quotient structure $\left\langle {{\Pi ^*}/R,{{B^\prime }_R},{\!\!{ \equiv ^\prime }_R}} \right\rangle $ is a model of ${{\cal E}_3}$ . Analogously, the members of ${\Pi ^*}$ may be regarded as micropoints, though they are not points in $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ . Moreover, the members of ${\Pi ^*}/R$ are the universes of the models of ${{\cal E}_3}$ contained in $\Pi $ , if $\Pi $ is understood as in Theorem 1, and they coincide with the members of $\Pi $ , if $\Pi $ is understood as in Theorem ${1^\dagger }$ . Accordingly, if $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ is understood as in Theorem ${1^\dagger }$ , then $\left\langle {{\Pi ^*}/R,{{B^\prime }_R},{\!\!{ \equiv ^\prime }_R}} \right\rangle $ is not merely isomorphic to $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ , it is identical to $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ .

In his mathematico-philosophical correspondence with Frege on the nature of the Euclidean axioms, Hilbert argued that the axioms constitute “only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements [such as points] can be thought of in any way one likes…provided only that the requisite axioms are satisfied” (December 12, [1899] Reference Hilbert, Gabriel, Hermes, Kambartel, Thiel, Veraart, Mcguinness and Kaal1980, 40–41). To illustrate this he notes: “If in speaking of my points I think of some system of things, e.g. the system: love, law, chimney-sweep…and then assume all my axioms as relations between these things, then my propositions, e.g. Pythagoras’ theorem, are also valid for these things” (December 29, Reference Hilbert1899, 40). This view, which was championed by Hilbert and a number of other late 19th- and early 20th-century geometers, has emerged as the standard construal of these matters of our day. Be that as it may, following longstanding tradition, points are still typically envisioned as unextended locations, or as locations having “no length, breadth or thickness” as Veblen ([1911] Reference Veblen and Young1955, 4) expressed it. The purpose of this paper has been to demonstrate that, longstanding tradition aside, the purported unextended nature of points is not implied by the axioms of ${{\cal E}_3}$ . Models of ${{\cal E}_3}$ whose universes consist of extended points do indeed exist, though one has to go outside the Euclidean model to recognize and characterize their extended nature.

On occasion it has been suggested to the author that the constructions of models of ${{\cal E}_3}$ with extended points presented above are simply the workings out of the details of instances of Hilbert’s quip about love, law, and chimney-sweeps^{Footnote 8} or his less well developed quip of 1882 where he maintains: “one must be able to say ‘tables, chairs, beer-mugs’ each time in place of ‘points, lines, planes’ ”(Blumenthal Reference Blumenthal1935, 402–3). While this was certainly not our intention, there is an inkling of truth in this contention, but only an inkling. As we noted in the Introduction, and as our models make clear, the extended points in our models are nondegenerate convex regions of models of paradigmatic extended geometrical spaces and are, as such, natural models of extended spatial regions—spatial regions which in principle might be nondegenerate regions of physical space. The same cannot be said of tables, love, or the like. As such, unlike the hypothesized playful models in Hilbert’s quips, our models go well beyond simply pointing out that points in a model of ${{\cal E}_3}$ may have unintended interpretations; they challenge the deep-seated and longstanding geometric shibboleth that points are necessarily unextended and as such are necessarily devoid of internal structure.

Nevertheless, could it be that our models of ${{\cal E}_3}$ with extended points, particularly the first where the extended points are themselves models of ${{\cal E}_3}$ , may exist only in a formally stipulated sense; that, for example, we lack any geometric or spatial conception of what constitutes the “betweenness” and “equidistance” between such extended points?^{Footnote 9} We believe the answer is no, though for the sake of space we will only touch on the matter here, leaving a more detailed historical and mathematico-philosophical response for another occasion.

To help motivate our negative response it will be useful to first consider a non-Archimedean model M of ${{\cal P}_3}$ having characteristic ordered field $F_{oe}^l$ containing an isomorphic copy ${\mathbb{R}_{oe}}$ of $\mathbb{R}$ , $oo$ and $oe$ being the zero and unit segments of $F_{oe}^l$ and hence ${\mathbb{R}_{oe}}$ (see sections 2, 3, and 5). Then M is isomorphic to ${C_3}(F_{oe}^l)$ , the latter of which contains a subspace ${C_3}({\mathbb{R}_{oe}})$ that is isomorphic to ${C_3}(\mathbb{R})$ . Moreover, since M is non-Archimedean, there are nondegenerate segments $oe^\prime $ and $oe^{\prime \prime} $ of l such that $oe^\prime $ is infinitesimal relative to $oe$ and $oe^{\prime \prime} $ is infinite relative to $oe$ . Furthermore, insofar as the characteristic ordered field of a model of ${{\cal P}_3}$ is (to within isomorphism) independent of the choice of unit segment, for each such pair $oe^\prime $ and $oe^{\prime \prime} $ there are isomorphic copies ${C_3}(F_{oe^\prime }^l)$ and ${C_3}(F_{oe^{\prime \prime} }^l)$ of M, whose characteristic ordered fields $F_{oe^\prime }^l$ and $F_{oe^{\prime \prime} }^l$ , respectively, contain isomorphic copies ${\mathbb{R}_{oe^\prime }}$ and ${\mathbb{R}_{oe^{\prime \prime} }}$ of $\mathbb{R}$ . Analogously, ${C_3}(F_{oe^\prime }^l)$ and ${C_3}(F_{oe^{\prime \prime} }^l)$ , respectively, contain subspaces ${C_3}({\mathbb{R}_{oe^\prime }})$ and ${C_3}({\mathbb{R}_{oe^{\prime \prime} }})$ , which are isomorphic to ${C_3}(\mathbb{R})$ . Thus, despite the fact that the nondegenerate segments in ${\mathbb{R}_{oe^\prime }}$ are infinitesimal relative to the nondegenerate segments in ${\mathbb{R}_{oe}}$ , which in turn are infinitesimal relative to the nondegenerate segments in ${\mathbb{R}_{oe^{\prime \prime} }}$ ,^{Footnote 10} the ordered fields ${\mathbb{R}_{oe}}$ , ${\mathbb{R}_{oe^\prime }}$ , and ${\mathbb{R}_{oe^{\prime \prime} }}$ are all isomorphic to $\mathbb{R}$ , and likewise for their corresponding Cartesian spaces ${C_3}({\mathbb{R}_{oe}})$ , ${C_3}({\mathbb{R}_{oe^\prime }})$ , and ${C_3}({\mathbb{R}_{oe^{\prime \prime} }})$ with respect to ${C_3}(\mathbb{R})$ .

While the above excursion is directed to non-Archimedean structures, it brings into focus the underappreciated fact that if M and $M^\prime $ are models of ${{\cal E}_3}$ , it is logically possible for the nondegenerate segments of M to be infinitesimal relative to the nondegenerate segments of $M^\prime $ . Accordingly, there is no logical difficulty in envisioning that the expansion $\left\langle {\Pi ,B,\!\!\equiv\!,(\left\langle {\mathbb{R}_x^3,{B_x},\!\!{\equiv \!_x}} \right\rangle {)_{x \in {\mathbb{R}^3}}}} \right\rangle $ of our first model of ${{\cal E}_3}$ models a hypothesized enriched Newtonian space in which the nondegenerate macrosegments—the nondegenerate segments modeled by the nondegenerate segments of $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ —are finite relative to the standard meter, and the nondegererate microsegments—the nondegenerate segments modeled by the nondegenerate segments of the $\left\langle {\mathbb{R}_x^3,{B_x},\!\!{ \equiv\! _x}} \right\rangle $ ’s—are infinitesimal relative to the standard meter. Indeed, whereas Hamlet figuratively proclaimed “I could be bounded in a nutshell and count myself a king of infinite space,” a denizen of the hypothesized microspace modeled by $\left\langle {\mathbb{R}_x^3,{B_x},\!\!{ \equiv \!_x}} \right\rangle $ could literally make the same claim with “nutshell” replaced by “point of (enriched) Newtonian space”! Of course, in this scenario the denizen of the microspace would be infinitesimal relative to the standard meter, despite the fact that the points of the microspace, together with the betweenness and equidistance relations defined thereon, model all of ${{\cal E}_3}$ . In this setting there is plainly no reason to believe the spatial betweenness and equidistance relations defined on the macropoints—the points of the enriched Newtonian space—would be other than the familiar ones. Indeed, it is difficult to see why this would not be the case whether the nondegenerate segments of the $\left\langle {\mathbb{R}_x^3,{B_x},\!\!{ \equiv _x}} \right\rangle $ ’s model segments that are finite, infinite, or infinitesimal relative to the standard meter; after all, in each such case the segments whose sole members are the $\left\langle {\mathbb{R}_x^3,{B_x},\!\!{ \equiv _x}} \right\rangle $ ’s would be the degenerate segments of $\left\langle {\Pi ,B,\!\!\equiv } \right\rangle $ .^{Footnote 11}

Besides the models of ${{\cal E}_3}$ with extended points constructed above, there is a wealth of other types of such models, including the aforementioned ones due to Huntington (see the Appendix). We will bring the main body of the paper to a close by drawing attention to a rather distinctive one, motivated by its inherent historical and mathematico-philosophical interest.

Since the time of Euclid it has been widely held that atomistic space and continuous space are incompatible (e.g., Hellman and Shapiro Reference Hellman and Shapiro2018, 190). After all, lines of atomistic space have been assumed to be discretely ordered—one space atom immediately following another—rather than densely ordered. However, if in our first construction of a model of ${{\cal E}_3}$ one replaces the copies of models of ${{\cal E}_3}$ indexed over ${\mathbb{R}^3}$ with copies of extended indivisible spatial atoms indexed over ${\mathbb{R}^3}$ , one obtains a model of ${{\cal E}_3}$ whose points are extended indivisible spatial atoms. That is, one obtains a three-dimensional Euclidean continuum of extended indivisible spatial atoms!