David Peat

On Incompleteness 41mathematics and made statements about mathematics was not itselfmathematics, but metamathematics, a system that lies outside mathematicsbut is used to describe it.Gödel’s stroke of genius was to discover a way of remaining withinmathematics by creating a symbolic system (within mathematics) thatrefers to itself and is therefore capable of making statements aboutitself—even to the point of demonstrating, or failing to demonstrate,its own consistency.The details of Gödel’s proof lie beyond the scope of this book—some hints are given in the Appendix. In essence, Gödel began by givingevery symbol a number. And of course numbers very naturally fallwithin the province of mathematics—they are not in the field ofmetamathematics. By combining these numbers in a special way, heshowed that every line of a proof could also be given a unique number.Every line of mathematics is defined by its own unique number. A persongiven that number can unpack it and write down that particularmathematical line.Next, every theorem—along with all the lines of its proof—is alsogiven a unique identifying number. Moreover, a statement about mathematics,a metastatement if you like, also has a number, and being anumber it is at the same time a part of arithmetic. Gödel was finallyable to arrive at numbers for statements such as “this true statement isnot demonstrable,” or “this statement is true” and “the negation of thisstatement is true.” In this way he was able to show that perfectly validnumbers in arithmetic correspond to statements like “this true statementis not demonstrable.” Thus Gödel was able to demonstrate thattrue statements exist that cannot be proved: in other words, that mathematicsis incomplete.What’s more, there are numbers in his system, that is, true statements,that correspond to “ this statement is true” and “the negation ofthis statement is true.” This means that inconsistencies also exist withinmathematics.Gödel had shown that mathematics is both incomplete andinconsistent. Mathematics must be incomplete because there will alwaysexist mathematical truths that can’t be demonstrated. Truths existin mathematics that do not follow from any axiom or theorem.