Disputes and paradoxes | Tangente

Richard's paradox ----------------------

Take the twenty-six letters of the alphabet and list, in alphabetical order, all strings of one letter, two letters, three letters, and so on. Everything that can be written appears in the resulting list. Cross out all strings that are not definitions of numbers (Jules Richard speaks of real numbers); let u₁ denote the first number on this list, u₂ the second, and so on. Thus, all numbers that can be defined using finitely many words form a countable set. Let E be the set of strings obtained in this way.

Richard then claims that we can construct a number that does not belong to this set. Let G be the following string of letters: "Let p be the nth decimal digit of the nth number in the set E; form a number N whose integer part is 0 and whose nth decimal digit is p+1 if p is neither 8 nor 9, and 1 otherwise." This number N does not belong to the set E, since if it were equal to, say, *u_n , its n*^th digit would be the n^th decimal digit of *u_n *, which it is not. Yet N is defined by the words in G—that is, by finitely many words—so it ought to belong to the set E!

Russell and Frege's dismay ------------------------------------