As a mathematician, Kurt Gödel produced a
number of important works which are difficult to understand
without some kind of editorial guidance or commentary. One
excellent book which deals with his incompleteness theorems (as
well as a score of other topics) is Douglas R. Hofstadter’s
Gödel, Escher, Bach: An Eternal Golden Braid.
Torkel Franzen’s Gödel’s Theorem: An
Incomplete Guide to Its Use and Abuse is a good text
explaining Gödel to non-mathematicians, and also follow’s
Gödel’s influence on modern science and thinking.
Gödel’s paper on Einstein’s field equations is the
subject of two books by Palle Yourgrau, Gödel Meets
Einstein and A World Without Time.
In German:
1931, "Über formal unentscheidbare Sätze der Principia
Mathematica und verwandter Systeme," Monatshefte für
Mathematik und Physik 38: 173-98.
1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger
Akademie der Wissenschaften Wien 69: 65-66.
In English:
1940. The Consistency of the Axiom of Choice and of the
Generalized Continuum Hypothesis with the Axioms of Set Theory.
Princeton University Press.
1947. "What is Cantor’s continuum problem?" The
American Mathematical Monthly 54: 515-25. Revised version in Paul
Benacerraf and Hilary Putnam, eds., 1984 (1964). Philosophy of
Mathematics: Selected Readings. Cambridge Univ. Press:
470-85.
In English translation:
Kurt Gödel, 1992. On Formally Undecidable Propositions Of
Principia Mathematica And Related Systems, tr. B. Meltzer, with a
comprehensive introduction by Richard Braithwaite. Dover reprint
of the 1962 Basic Books edition.
Kurt Gödel, 2000.
http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf
On Formally Undecidable Propositions Of Principia Mathematica And
Related Systems, tr. Martin Hirzel
Jean van Heijenoort, 1967. A Source Book in Mathematical Logic,
1879-1931. Harvard Univ. Press.
1930. "The completeness of the axioms of the functional
calculus of logic," 582-91.
1930. "Some metamathematical results on completeness and
consistency," 595-96. Abstract to (1931).
1931. "On formally undecidable propositions of Principia
Mathematica and related systems," 596-616.
1931a. "On completeness and consistency," 616-17.
Collected Works: Oxford University Press: New York.
Editor-in-chief: Solomon Feferman.
Volume I: Publications 1929-1936 ISBN 0195039645
Volume II: Publications 1938-1974 ISBN 0195039726
Volume III: Unpublished Essays and Lectures ISBN 0195072553
Volume IV: Correspondence, A-G ISBN 0198500734
Volume V: Correspondence, H-Z ISBN 0198500750
