Revised Edition, New York University Press, New York and London 2001
Edited and with a New Foreword by Douglas R. Hofstadter
en.wikipedia.org/wiki/Gödel’s_incompleteness_theorems
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One of the axioms Euclid used is systematizing geometry has to do with parallels. The axiom he adopted is logically equivalent to (though not identical with) the assumption that through a point outside a given line only one parallel to the line can be drawn. For various reasons, this axiom did not appear "self-evident" to the ancients. They sought, therefore, to deduce it from the other Euclidean axioms, which they regarded as clearly self evident. Can such a proof of the parallel axiom be given? Generations of mathematicians struggled with this question, without avail. But repeated failure to construct a proof does not mean that none can be found any more than repeated failure to find a cure for the common cold establishes beyond doubt that humanity will forever suffer from running noses. It was not until the nineteenth century, chiefly through the work of Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated. This outcome was of the greatest intellectual importance. In the first place, it called attention in a most impressive way to the fact that a proof can be given of the impossibility of proving certain propositions within a given system. As we shall see, Gödel’s paper is a proof of the impossibility of formally demonstrating certain important propositions in number theory.
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Gödel showed (i) how to construct a formula G of PM that represents the meta-mathematical statement: ‘The formula G is not demonstrable using the rules of PM’. This formula 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 likewise assosiated with a certain number g – namely, its Gödel number – and G is constructed that it says ‘The formula that has Gödel number g 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 the 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, then PM is not consistent. Accordingly, if PM is consistent, neither G nor ~G can be formally derivable from the axioms. In short, if PM is consistent, then G is a formally undecidable formula.
Gödel then showed (iii) that, though G is not formally demonstrable, it nevertheless is a true arithmetical formula. G is true in the sense that it claims that a certain arithmetical property defined by Gödel is possessed by no integer – and indeed, no integer possesses the property, Gödel shows.
Step (iv) is the realization that since G is both true and formally undecidable (within PM), PM must be incomplete. In other words, we cannot deduce all arithmetical truths from the axioms and rules of PM. Moreover, Gödel established that PM is essentially incomplete: even if PM were augmented by additional axioms (or rules) so that the true formula G could be formally derived within the enhanced calculus, then another true formula G’ could be constructed in a precisely analogous manner, and G’ would be formally undecidable inside the enhanced calculus. Needless to say, further enhancement of the already-enhanced calculus, so as to allow derivation of G’, would merely lead to yet another formula G” undecidable within the doubly augmented system – and so on, ad infinitum. This is the meaning of "essentially incomplete".
In step (v), Gödel described how to construct a formula A of PM that represents the meta-mathematical statement: ‘PM is consistent’; and he showed that the formula ‘A ⊃ G’ is formally demonstrable inside PM. Finally, he showed that the formula A is not demonstrable inside PM. From this it follows that the consistency of PM cannot be established by any chain of logical reasoning that can be mirrored within the formal reasoning system that PM itself constitues.
(PM – Principia Mathematica, Richard’s paradox, Metamathematics – jh)
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