David Peat

Appendix: Gödel’s Theorem 219It was at this point that Gödel proposed using prime numbers. (Aprime number, such as 7, 11, 13, 23, etc., cannot be factored into othernumbers, whereas a nonprime such as 12 can be factored into 2 × 6,and 3 × 4.)Thus the new Gödel number for 2 + 2 = 4 now becomes (using thesequence 9, 1, 9, 4, 11):2 9 × 3 1 × 5 9 × 11 4 × 13 11This (very large) number is unique and is shared by no other formula.Thus could Gödel express every symbol and every formula in mathematicsby a unique Gödel number.Step 3What applies to a single formula also applies to a theorem and its proof.Simply assign numbers to each of the symbols and to each line in theproof. Now work out the Gödel number for this proof Y. We can nowsay that the theorem with Gödel number X has a proof with Gödelnumber Y. This is written down as Dem(Y, X).But it may also be the case that some sequence Y does not provethe truth of the theorem X. Gödel wrote this as ~Dem(Y,X). Clearly ifboth Dem(Y,X) and ~Dem(Y,X) could be shown to coexist then mathematicswould be contradictory.Step 4We have come quite a distance in the argument for we can now reduceall the theorems of mathematics and all their proofs to a series of Gödelnumbers. Yet we still have not shown how a logical system can actuallyrefer to itself. One further turn of the logical screw is necessary.Suppose we take one of the basic facts of mathematics—that thenumbers go on forever so that every number y has a successor x. Mathematicianslike to phrase this in the following way, “There exists somenumber x, such that x is the successor of y.” Written as a formula thisreads