David Peat

218 From Certainty to UncertaintyStep 1Let us begin by arranging in a row all the symbols used in mathematics,along with all the symbols of logic and all the numbers. The listbelow is a great simplification but serves to explain the general idea ofhow things are going to work.+ – × = x y 0 1 2 3 4 5 6Just as in the counting game place numbers below this row+ – × = x y 0 1 2 3 4 5 61 2 3 4 5 6 7 8 9 10 11 12 13If you give me the Gödel number 4, I now know that it stands forthe equals sign, while the Gödel number 10 stands for the number 3.Step 2Not only can individual symbols and numbers be given a Gödel number,but an entire formula can be reduced to a number. Take as anexample the formula 2 + 2 = 4. This involves the Gödel numbers 9(which stands for the number 2), 1 (for the + sign), 4 (for equals), and11 (for 4). We now add these numbers to get a new single Gödel number34:9 + 1 + 9 + 4 + 11 = 34Thus the Gödel number 34 stands for the formula 2 + 2 = 4.But now we run into a serious problem, for 34 is also the numberfor the formula 3 + 1 = 4. At this stage what we have been taking forGödel numbers are not unique and, when we deal with more complicatedformulas, one Gödel number can stand for a number of differentformulas. Clearly this will not suffice to define mathematical formulasin a unique way.