> I can't see how to guarantee that this generated number wasn't already in the set. The trick here is the diagonal argument [1]. The set (S) contains all rea

> I can't see how to guarantee that this generated number wasn't already in the set. The trick here is the diagonal argument [1]. The set (S) contains all real numbers that can be described by a valid French sentence (enumerating French sentences in some fixed order). For example we could come up with an enumeration such that the first few elements are: S_0 = .7139847654 S_1 = .111111111111 S_2 = .93939 S_3 = .313331333133 repeating ... We can construct a diagonal number r such that the ith digit of r differs from the ith digit of S_i for all S_i in S: r = 0.8204... (8 = 1 + S_0[0], 2 = 1 + S_1[1], 0 = S_2[2] + 1, etc) For any element S_i in S; r and S_i differ in at least one digit (because we constructed it that way); which is why r is guaranteed to not be in S. (But this argument can be translated to French, hence the paradox!) [1] https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument