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Metric postulates for time geometry

Geometry was developed by the ancient Greeks in the language of length, but it is an abstraction that may be applied to anything that conforms to its definitions and axioms. Here we apply it to duration. We will use Brossard’s “Metric Postulates for Space Geometry” [American Mathematical Monthly, Vol. 74, No. 7, Aug.-Sep., 1967, pp. 777-788], which generalizes the plane metric geometry of Birkhoff to three dimensions. First, Brossard:

Primitive notions. Points are abstract undefined objects. Primitive terms are: point, distance, line, ray or half-line, half-bundle of rays, and angular measure. The set of all points will be denoted by the letter S and some subsets of S are called lines. The plane as well as the three-dimensional space shall not be taken as primitive terms but will be constructed.

Axioms on points, lines, and distance. The axioms on the points and on the lines are:

E1. There exist at least two points in S.

E2. A line contains at least two points.

E3. Through two distinct points there is one and only one line.

E4. There exist points not all on the same line.

A set of points is said to be collinear if this set is a subset of a line. Two sets are collinear if the union of these sets is collinear. The axioms on distance are:

D1. If A and B are points, then d(AB) is a nonnegative real number.

D2. For points A and B, d(AB) = 0 if and only if A = B.

D3. If A and B are points, then d(AB) = d(BA).

For distinct points A, B, C; B is between A and C if and only if d(AB) + d(BC) = d(AC). If B is between A and C we write [A, B, C]. If AB, the class of points P between A and B is the interval AB. If AO, the class of points PO such that d(AP) = | d(OA) − d(OP) | is the ray OA (or half-line OA). O is the end-point of the ray OA. We have furthermore the two following axioms concerning the distance.

D4. Three distinct points belong to a line if and only if one of them is between the other two.

D5. On each ray OA and for each positive real number β there exists a point B such that d(OB) = β.

The axioms on angles, half-bundles, and the continuity axiom. An angle is an ordered couple of rays with the same end-point. The rays are the sides of the angle and the common end-point is the vertex of the angle. An angle is flat in case the sides are distinct and belong to a line. We shall introduce the measure ∠rs of an angle rs with the axioms A1, A2, A3. The measure of an angle is intuitively conceived as the number obtained with the use of a protractor. If AO and BO, we write ∠AOB for ∠(ray OA)(ray OB).

A1. If rs is an angle, then ∠rs is a real number such that 0 ≤ ∠rs ≤ 180.

A2. For an angle rs, ∠rs = 0 if and only if r = s.

A3. If rs is an angle, then ∠rs = ∠sr.

A half-bundle ab, where a and b are rays with the same end-point not belonging to a line, is the class of rays x such that x ≠a and ∠bx = | ∠ab ∠ax |. The common end-point of the rays is the vertex of the half-bundle. For the half-bundle ab we use the notation HBab. A ray b is between the rays a and c if and only if a, b, c have a common end-point, no two of them belong to a line, and ∠ab + ∠bc = ∠ac; if b is between a and c we write [a, b, c]. With those notations we have that x ∈ HBab if and only if [a, b, x] or [a, x, b] or x = b.

The axioms on the half-bundles are:

A4. Three distinct rays a, b, c belong to a half-bundle if and only if one of them is between the other two.

A5. If β is a real number between 0 and 180, then there exists on each half-bundle HBoa one ray b such that ∠ob = β.

We shall now add the so-called continuity axiom.

Continuity Axiom. If [a, b, c] and if A, C are points on rays a and c respectively, then there exists a point B on ray b such that [A, B, C]. Conversely, if a and c are noncollinear rays with the same end-point O, Aa, Cc, and if B is such that [A, B, C] then the ray OB = b is between the rays a and c (i.e. [a, b, c]).


What follows is the above abstraction with terms that suggest duration instead of length.

Primitive notions. Timepoint are abstract undefined objects. Primitive terms are: timepoint, distance, timeline, ray or half-line, half-bundle of rays, and turn measure. The set of all timepoints will be denoted by the letter S and some subsets of S are called timelines. The timeplane as well as three-dimensional time shall not be taken as primitive terms but will be constructed.

Axioms on timepoints, timelines, and distance. The axioms on the timepoints and timelines are:

E1. There exist at least two timepoints in S.

E2. A timeline contains at least two timepoints.

E3. Through two distinct timepoints there is one and only one timeline.

E4. There exist timepoints not all on the same timeline.

A set of timepoints is said to be collinear if this set is a subset of a timeline. Two sets are collinear if the union of these sets is collinear. The axioms on distance are:

D1. If A and B are timepoints, then d(AB) is a nonnegative real number.

D2. For timepoints A and B, d(AB) = 0 if and only if A = B.

D3. If A and B are timepoints, then d(AB) = d(BA).

For distinct timepoints A, B, C; B is between A and C if and only if d(AB) + d(BC) = d(AC). If B is between A and C we write [A, B, C]. If AB, the class of timepoints P between A and B is the interval AB. If AO, the class of timepoints PO such that d(AP) = | d(OA) − d(OP) | is the ray OA (or half-line OA). O is the end-timepoint of the ray OA. We have furthermore the two following axioms concerning the distance.

D4. Three distinct timepoints belong to a timeline if and only if one of them is between the other two.

D5. On each ray OA and for each positive real number β there exists an timepoint B such that d(OB) = β.

The axioms on turns, half-bundles, and the continuity axiom. A turn is an ordered couple of rays with the same end-timepoint. The rays are the sides of the turn and the common end-timepoint is the vertex of the turn. A turn is flat in case the sides are distinct and belong to a timeline. We shall introduce the measure ∠rs of a turn rs with the axioms A1, A2, A3. The measure of a turn is intuitively conceived as the number obtained with the use of a dial. If AO and BO, we write ∠AOB for ∠(ray OA)(ray OB).

A1. If rs is a turn, then ∠rs is a real number such that 0 ≤ ∠rs ≤ 180.

A2. For a turn rs, ∠rs = 0 if and only if r = s.

A3. If rs is a turn, then ∠rs = ∠sr.

A half-bundle ab, where a and b are rays with the same end-timepoint not belonging to a timeline, is the class of rays x such that x ≠a and ∠bx = | ∠ab ∠ax |. The common end-timepoint of the rays is the vertex of the half-bundle. For the half-bundle ab we use the notation HBab. A ray b is between the rays a and c if and only if a, b, c have a common end-timepoint, no two of them belong to a timeline, and ∠ab + ∠bc = ∠ac; if b is between a and c we write [a, b, c]. With those notations we have that x ∈ HBab if and only if [a, b, x] or [a, x, b] or x = b.

The axioms on the half-bundles are:

A4. Three distinct rays a, b, c belong to a half-bundle if and only if one of them is between the other two.

A5. If β is a real number between 0 and 180, then there exists on each half-bundle HBoa one ray b such that ∠ob = β.

We shall now add the so-called continuity axiom.

Continuity Axiom. If [a, b, c] and if A, C are timepoints on rays a and c respectively, then there exists an timepoint B on ray b such that [A, B, C]. Conversely, if a and c are noncollinear rays with the same end-timepoint O, Aa, Cc, and if B is such that [A, B, C] then the ray OB = b is between the rays a and c (i.e. [a, b, c]).

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