 # Cardinality

To understand the assertion that time is a continuum, we need to understand continuous sets: therefore we need to develop some further set theoretic concepts.

Two sets A and B are equinumerous, written A ~ B, if there exists a bijection from A to B. This means every element of A can be paired with one and only one element of B and vice-versa. This leads to the concept of cardinality, a statement of how ‘big’ a given set is.

For any set A, we assign to it a cardinal number called the cardinality of A and denoted #A. The cardinality of a set is a measure of how many elements there are in that set. Cardinal numbers come in two varieties: finite cardinal numbers are equivalent to ordinary whole numbers, that is the non-negative integers, whilst infinite cardinal numbers are altogether stranger beasts, the subject of much mathematical debate in the nineteenth century.

# The finite cardinals

To construct the set of finite cardinals, we start with the empty set 0. Given that the empty set is defined to be a set with no elements at all, we define the cardinality of the empty set to be the cardinal number 0, corresponding to the integer zero, that is #0 = 0.

Now define the set 0 = {0}. This is not the empty set: it contains one element, which happens to be the empty set. Therefore we define #0 = 1, corresponding to the familiar integer one.

Next, defining the set 0 = {0,0}, we write #0 = 2.

Next, defining the set 0 = {0,0, 0}, we write #0 = 3.

Continuing this process, the general extension is 0[n] = {0,0,. . . 0[n-1]}, with #0[n] = n.

A set A has finite cardinality n if A is equinumerous with 0[n], that is where the ‘if and only if’ symbol ^ is the mathematician’s notation for ‘the right- hand side is always true if the left-hand side is true, and vice-versa’.