Gödel's Incompleteness Theorem

Informally, Gödel's incompleteness theorem states that all consistent axiomatic formulations of Number Theory includeundecidable propositions (Hofstadter 1989). This is sometimes called Gödel's first incompleteness theorem, and answersin the negative Hilbert's Problem asking whether mathematics is ``complete'' (in thesense that every statement in the language of Number Theory can be either proved or disproved). Formally, Gödel'stheorem states, ``To every -consistent recursive class of Formulas, there correspondrecursive class-signs such that neither ( Gen ) nor Neg( Gen ) belongs to Flg(), where is theFree Variable of '' (Gödel 1931).

A statement sometimes known as Gödel's second incompleteness theorem states that if Number Theory is consistent, thena proof of this fact does not exist using the methods of first-order Predicate Calculus. Statedmore colloquially, any formal system that is interesting enough to formulate its own consistency can prove its ownconsistency Iff it is inconsistent.

Gerhard Gentzen showed that the consistency and completeness of arithmetic can be proved if ``transfinite'' induction is used.However, this approach does not allow proof of the consistency of all mathematics.

References

Barrow, J. D. Pi in the Sky: Counting, Thinking, and Being. Oxford, England: Clarendon Press, p. 121, 1992.

Gödel, K. ``Über Formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme, I.'' Monatshefte für Math. u. Physik 38, 173-198, 1931.

Gödel, K. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. New York: Dover, 1992.

Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 17, 1989.

Kolata, G. ``Does Gödel's Theorem Matter to Mathematics?'' Science 218, 779-780, 1982.

Smullyan, R. M. Gödel's Incompleteness Theorems. New York: Oxford University Press, 1992.

Whitehead, A. N. and Russell, B. Principia Mathematica. New York: Cambridge University Press, 1927.

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