David Peat

220 From Certainty to Uncertainty(∃x)(x = sy) (1)The number y has, as we have seen above, the Gödel number 6. Inaddition, the entire formula has its own Gödel number that can becalculated like any other Gödel number. Call the number of the formulaabove M.In turn we can put this number M back into the formula in placeof the number 6 (that is, the variable y).(∃x)(x = sM) (2)This formula says “there exists some number x such that it is the successorto the number M.” Admittedly M is a Gödel number but as anumber in arithmetic it is no different from the number 6 (whichstands for y).As with every other formula in mathematics, line (2) has a Gödelnumber that can be calculated. But there is a second way to calculatethat number. We can calculate it in such a way that metamathematicsbegin to mirror and reflect each other.Suppose we write down the statement: “The formula obtainedfrom the formula whose number is ‘M ’ when you substitute the numberM for the variable with number 6.”—Statement (a)Statement (a) is unambiguous. It’s a statement that can be writtendown in symbolic form so that its Gödel number, N, can be calculated.In other words the statement with Gödel number N (Statement a) andline (2) are mirror images of each other—mathematics and metamathematicsnow reflect each other within the same system.Having achieved this result—mirroring metamathematical statementswithin mathematics itself—Gödel could go on to construct thestatement: “The formula with Gödel number Z is not demonstrable.”—Statement (b)In other words Gödel had constructed a statement of the type “Iam not demonstrable” or “I cannot be proved.”Gödel also added two final steps to the argument. First, he showedthat although Statement (b) cannot be demonstrated within his sys-