Gödel showed (i) how to construct an arithmetical formula G that represents the meta-mathematical statement: ‘The formula G is not demonstrable’. This formula G thus ostensibly says of itself that it is not demonstrable. Up to a point, G is constructed analogously to the Richard Paradox. In that Paradox, the expression ‘Richardian’ is associated with a certain number n, and the sentence ‘n is Richardian’ is constructed. In Gödel’s argument, the formula G is also associated with a certain number h, and is so constructed that it corresponds to the statement: ‘The formula with the associated number h is not demonstrable’. But (ii) Gödel also showed that G is demonstrable if, and only if, its formal negation ~G is demonstrable. This step in the argument is again analogous to a step in the Richard Paradox, in which it is proved that n is Richardian if, and only if, n is not Richardian. However, if a formula and its own negation are both formally demonstrable, the arithmetical calculus is not consistent. Accordingly, if the calculus is consistent, neither G nor ~ G is formally derivable from the axioms of arithmetic. Therefore, if arithmetic is consistent, G is a formally undecidable formula. Gödel then proved (iii) that, though G is not formally demonstrable, it nevertheless is a true arithmetical formula. It is true in the sense that it asserts that every integer possesses a certain arithmetical property, which can be exactly defined and is exhibited by whatever integer is examined, (iv) Since G is both true and formally undecidable, the axioms of arithmetic are incomplete. In other words, we cannot deduce all arithmetical truths from the axioms. Moreover, Gödel established that arithmetic is essentially incomplete: even if additional axioms were assumed so that the true formula G could be formally derived from the augmented set, another true but formally undecidable formula could be constructed, (v) Next, Gödel described how to construct an arithmetical formula A that represents the meta-mathematical statement: ‘Arithmetic is consistent’; and he proved that the formula ‘A ⊃ G’ is formally demonstrable. Finally, he showed that the formula A is not demonstrable. From this it follows that the consistency of arithmetic cannot be established by an argument that can be represented in the formal arithmetical calculus.
(From Nagel and Newman‘s “Gödel’s Proof”)