Lesson 5

Note that Cantor's diagonal method is also used in this proof, although it is not entirely clear which diagonal and which table is being referenced. The table can be imagined as follows: let us assume that the rows are indexed by elements of \(X\), and the columns by elements of \(2^{X}\) (usually, we require the sets we are indexing rows or columns with to be countable, but here, no such guarantee has been made, though this is not a problem). Next, assume that there is a bijection between \(X\) and \(2^{X}\). This, in some sense, means that our table will be square. We will assume that the columns are indexed by the preimages of the row indices. In each cell of the table, we will write 0 or 1, depending on whether an element is in the set or not. For example, when \(X=\{a,c\}\), the following table would result.