Temporal ordering and time labels
We need two or more events to define time. Suppose we have observed two events A and B. Then according to our own sense of time we will generally feel entitled to say that one of three possible mutually exclusive temporal relationships exists between these events. We may say that either A is earlier than B, or A is later than B, or that A and B are simultaneous, that is, occur at the same time.
Sooner or later, we will need to write down our observations. This will require us to agree on some notation. Suppose we decide on the following minimal scheme: if event A has occurred before B we will write A < B, whilst if A has occurred later than B we write A > B. According to this convention, A < B means the same thing as B > A. If the events are simultaneous, we write A = B. At this point, we have not introduced real numbers at all: the symbols < and > are just notation for earlier than and later than.
There is one problem with the above notation. The simultaneity of two events does not mean that they are the same event: A = B is misleading notation for simultaneity. To get around this, we introduce the concept of an ordered set, discussed in Chapter 7. A well-known set with these properties is R, the set of real numbers. It is the ordered set most commonly used to model time. The procedure is to associate to each event A a unique real number tA, called the time of that event. The notions of later, earlier, and simultaneous between events are then encoded into the various possible ordering relationships between real numbers. For example, if event A is later than event B we write tA > tB whilst if A and B are simultaneous we write tA = tB. This latter equality then does not mean that A = B in the mathematical sense of equality [Howson, 1972].