Austrian-American mathematician who proved that, if you begin with any sufficiently strong consistent system of axioms, Eric Weisstein's World of Math there will always be statements within the system governed by those axioms Eric Weisstein's World of Math that can neither be proved or disproved on the basis of those axioms. Eric Weisstein's World of Math Hence, it in undecidable Eric Weisstein's World of Math on the basis of those axioms whether the system contains paradoxes. Eric Weisstein's World of Math The formal statement of this fact is known as Gödel's incompleteness theorem. Eric Weisstein's World of Math
Gödel also proved Gödel's completeness theorem, Eric Weisstein's World of Math which states that if T is a set of axioms Eric Weisstein's World of Math in a first-order language, and a statement p holds for any structure M satisfying T, then p can be formally deduced from T in some appropriately defined fashion.
Gödel showed that no contradiction would arise if the continuum hypothesis Eric Weisstein's World of Math were added to conventional Zermelo-Fraenkel set theory. Eric Weisstein's World of Math However, using a technique called forcing, Eric Weisstein's World of Math Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to set theory. Eric Weisstein's World of Math Together, Gödel's and Cohen's results established that the validity of the continuum hypothesis depends on the version of set theory Eric Weisstein's World of Math being used, and is therefore undecidable Eric Weisstein's World of Math (assuming the Zermelo-Fraenkel axioms Eric Weisstein's World of Math together with the axiom of choice Eric Weisstein's World of Math).
Additional biographies: MacTutor (St. Andrews)
References
Cohen, P. J. "The Independence of the Continuum Hypothesis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143-1148, 1963.
Cohen, P. J. "The Independence of the Continuum Hypothesis. II." Proc. Nat. Acad. Sci. U. S. A. 51, 105-110, 1964.
Dawson, J. W. Jr. Logical Dilemmas: The Life and Work of Kurt Gödel. New York: A. K. Peters, 1997.
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. New York: Dover, 1992.
Gödel, K. Collected Papers, Vol. 1: Publications 1929-1936. Oxford, England: Oxford University Press, 1986.
Gödel, K. Collected Papers, Vol. 2: Publications 1938-1974. Oxford, England: Oxford University Press, 1989.
Gödel, K. Collected Papers, Vol. 3: Unpublished Essays and Lectures. Oxford, England: Oxford University Press, 1995.
Heijenoort, J. van. From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1931. Cambridge, MA: Cambridge University Press, 1967.
Hoffman, P. The Man Who Loved Only Numbers. New York: Hyperion, 1998.
Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 15-19, 1989.
Rodriguez-Consuegra, F. A. Kurt Gödel: Unpublished Philosophical Essays. Boston, MA:: Birkhäuser, 1996.
