A self-referential sentence is a sentence that refers to itself. Examples are easy to find, such as "This sentence contains five words.", which also states a true property (P). The sentence would be just as self-referential if (P) were false. Property (P) can also be paradoxical—"neither true nor false," in a sense—as in the classic example: "This sentence is a lie." If this sentence really is a lie, then it is telling the truth, so it is not a lie. If it is telling the truth, then it is a lie. That is the paradox.
Self-reference is closely related to recursive definitions in mathematics, such as that of the factorial function fact, which satisfies the self-referential property fact(n) = n × fact(n – 1). This equality defines the factorial function, provided that fact(0) = 1 is added. A rigorous proof of this result uses induction. This example shows that a self-referential definition can be valid.
Successive approximations
The method of successive approximations solves an equation that can be written in the form f(x) = x by using a sequence (un)n≥0 defined by its first term u0 and the recurrence relation un+1 = f(un). For example, take the equation x2 – x – 2 = 0, which can be rewritten as x2 – 2 = x. Let f(x) = x2 – 2. Starting from u0 = 0, we obtain u1 = –2, u2 = 2, u3 = 2… Thus 2 is a fixed point of f and therefore a root of the equation under consideration. Starting from u0 = 1 gives another fixed point, –1. This yields both roots of the given equation.
Of course, this method does not work for every choice of the function f: certain assumptions are needed. Surprisingly, it also works beyond a purely numerical setting—for example, when searching for a self-referential sentence. More precisely, the task is to complete the following sentence with a number, written out in words, that makes it true:
