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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’.

 
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