Page 233

Location: Chapter 25, last lines of Introduction before Problem 1

It is

... For example, suppose \(X\) is the expression \(S\) and \(Y\) is the expression \(K\). The Gödel number of \(X\) is \(31\) and the Gödel number of \(Y\) is \(24\). The expression \(XY\) is \((SK)\) and its Gödel number is \(3124\), which is \(31 \star 24\). Now you can see the significance of the numerical operation of concatenation to the base \(10\).

It should become

... For example, suppose \(X\) is the expression \(S\) and \(Y\) is the expression \(K\). The Gödel number of \(X\) is \(1\) and the Gödel number of \(Y\) is \(2\). The expression \(XY\) is \((SK)\) and its Gödel number is \(3124\), which is \(3 \star 1 \star 2 \star 4\). Now you can see the significance of the numerical operation of concatenation to the base \(10\).

Short explanation

Throughout Chapter 25, parentheses, generally, appear to have been reinstated explicitly: since each symbol yields a digit in the expression's Gödel number, even "obvious" parentheses make a difference. Besides, in the formal definition of term a couple of pages earlier, it is said that "given any terms \(X\) and \(Y\) already constructed, we may form the new term \((XY)\)", the latter expression being explicitly parenthesized.

In view of this fact, it is better to explicitly observe that the symbols \(S\), \(K\) alone correspond to the numbers \(1\) and \(2\), the rest of the complete result \(3124\) coming from the compulsory parentheses.