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

Next, defining the set 0^{[2]} = {0,0^{[1]}}, we write #0^{[2]} = 2.

Next, defining the set 0^{[3]} = {0,0^{[1]}, 0^{[2]}}, we write #0^{[3]} = 3.

Continuing this process, the general extension is 0^{[n]} = {0,0^{[1]},. . . 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’.