Time

In GOL, time is divided in two basic entities: time boundaries and chronoids. Chronoids are time intervals, and they always have a duration. Time boundaries depend on chronoids: A chronoid has exactly two time boundaries, a left and a right time boundary. We denote chronoids with $chr(i)$, meaning $i$ is a chronoid, and time-boundaries with $tb(b)$, meaning $b$ is a time boundary.

We use the relations:

1. $bl(x,y) =_{def}$ $x$ is left boundary of the chronoid $y$
2. $br(x,y) =_{def}$ $x$ is right boundary of the chronoid $y$
3. $coinc(x,y) =_{def}$ the time boundaries $x$ and $y$ coincide

Defined is $b(x,y) =_{def} br(x,y)\lor bl(x,y)$.

Let us view the axioms of this theory as in stenzel1.

The first axiom asserts the existence of a chronoid.

Axiom 4.1  

$$\forall i (tb(i) \rightarrow \exists !! x \left( int(x) \land b(x,i) \right) )$$

The second axiom asserts the uniqueness of the left and right boundary of a chronoid.

Axiom 4.2  

$$\forall x \left( int(x) \rightarrow \exists !!u \left(bl(u,x) \land \exists !!v \left( br(v,x) \right) \right) \right)$$

The next axiom states that time boundaries and chronoids are disjunct.

Axiom 4.3  

$$\neg \exists x \left( int(x) \land tb(x) \right)$$

The next axiom asserts that time is infinite: Every left boundary of a chronoid coincides with the right boundary of another chronoid, and vice versa.

Axiom 4.4  

$$\forall x,u \left( bl(x,u) \rightarrow \exists y,v \left( br(y,v) \land coinc(x,y) \right) \right)$$

Axiom 4.5  

$$\forall x,u \left( br(x,u) \rightarrow \exists y,v \left( bl(y,v) \land coinc(x,y) \right) \right)$$

Chronoids may have internal structure.

Axiom 4.6  

$$\begin{split}\forall x,y,a,b,c,d \left( br(b,x) \land bl(c,y)... ...coinc(a,e) \land coinc(d,f) \right) \right) \right) \end{split}$$

The last axiom asserts the linearity of time.

Axiom 4.7  

$$\begin{split}\forall x,y \left( tb(x) \land tb(y) \land \neg ... ...land \ coinc(a,x) \land coinc(b,x) \right) \right) \end{split}$$

This theory of time and the theory introduced by allen1, using only one relation, the relation $meets$, are equivalent, as shown in stenzel1. Sometimes this model of time is called the ``glass continuum''.