Turing's Proof - Richard's Paradox

Richard's Paradox

In 1905 Jules Richard presented this profound paradox. Alan Turing's first proof constructs this paradox with his so-called computing machine and proves that this machine cannot answer a simple question: will this machine be able to determine if any computing machine (including itself) will become trapped in an unproductive "infinite loop" (i.e. it fails to continue its computation of the diagonal number).

A succinct definition of Richard's paradox is found in Whitehead and Russell's Principia Mathematica:

"Richard's paradox... is as follows. Consider all decimals that can be defined by means of a finite number of words; let E be the class of such decimals. Then E has terms; hence its members can be ordered as the 1st, 2nd, 3rd, ... Let N be a number defined as follows ; If the nth figure in the nth decimal is p, let the nth figure in N be p+1 (or 0, if p = 9). Then N is different from all the members of E, since, whatever finite value n may have, the nth figure in N is different from the nth figure in the nth of the decimals composing E, and therefore N is different from the nth decimal. Nevertheless we have defined N in a finite number of words and therefore N ought to be a member of E. Thus N both is and is not a member of E" (Principia Mathematica, 2nd edition 1927, p. 61).