Reference

All of Gödel’s published work, together with a large number of the unpublished material from the Nachlass, together with a selection of Gödel’s correspondence is published in Kurt Gödel, Collected Works, Volumes I-V. Secondary Sources Academic Tools Other Internet Resources Related Entries Gödel, Kurt: incompleteness theorems | Hilbert, David: program in the foundations of mathematics | Husserl, Edmund | Leibniz, Gottfried Wilhelm | mathematics, philosophy of: intuitionism | mathematics, philosophy of: Platonism | model theory | model theory: first-order | phenomenology | realism | set theory | set theory: continuum hypothesis | set theory: large cardinals and determinacy Acknowledgments Though van Heijenoort and Dreben (Dreben and van Heijenoort 1986) remark that “Throughout much of the 1920s it was not semantic completeness but the decision problem for quantificational validity, a problem originating from the work of Schröder and Löwenheim, that was the dominant concern in studying quantification theory” (examples of such results would include the decision procedure for the first order monadic predicate calculus due to Behmann, (Behmann 1922)), according to Gödel, the reasons that Skolem did not obtain the complete proof are different and philosophically important, having to do with the then dominant bias against semantics and against infinitary methods: The matter of Skolem’s contribution to the Completeness Theorem has been extensively discussed in van Atten and Kennedy 2009, as well as in van Atten 2005. As he remarked at the end of his 1947 “What is Cantor’s Continuum Hypothesis?” Gödel was compelled to this view of L by the Leibnizian[18] idea that, rather than the universe being “small,” that is, one with the minimum number of sets, it is more natural to think of the set theoretic universe as being as large as possible.[19]This idea would be reflected in his interest in maximality principles, i.e., principles which are meant to capture the intuitive idea that the universe of set theory is maximal in the sense that nothing can be added; and in his conviction that maximality principles would eventually settle statements like the CH. Skolem’s observation that categoricity fails for set theory because it has countable models is now known as the Skolem paradox.[8]The observation is strongly emphasized in Skolem’s paper, which is accordingly entitled ‘An Observation on the Axiomatic Foundations of Set Theory’ As he wrote in the conclusion of it, he had not pointed out the relativity in set theory already in 1915 because: … first, I have in the meantime been occupied with other problems; second, I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. 2.1 The Completeness Theorem The completeness question for the first order predicate calculus was stated precisely and in print for the first time in 1928 by Hilbert and Ackermann in their text Grundzüge der theoretischen Logik (Hilbert and Ackermann 1928), a text with which Gödel would have been quite familiar.[6] The question Hilbert and Ackermann pose is whether a certain explicitly given axiom system for the first order predicate calculus “…is complete in the sense that from it all logical formulas that are correct for each domain of individuals can be derived…” (van Heijenoort 1967, p. 48).

The volume has three parts: Part I, "Historical Context: Gödel's Contributions and Accomplishments," contains ten essays; Part II, "A Wider Vision: The Interdisciplinary, Philosophical, and Theological Implications of Gödel's Work," contains six essays; Part III, "New Frontiers: Beyond Gödel's Work in Mathematics and Symbolic Logic ...

Gödel’s theorems In his doctoral thesis, “Über die Vollständigkeit des Logikkalküls” (“On the Completeness of the Calculus of Logic”), published in a slightly shortened form in 1930, Gödel proved one of the most important logical results of the century—indeed, of all time—namely, the completeness theorem, which established that classical first-order logic, or predicate calculus, is complete in the sense that all of the first-order logical truths can be proved in standard first-order proof systems. In 1940, only months after he arrived in Princeton, Gödel published another classic mathematical paper, “Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory,” which proved that the axiom of choice and the continuum hypothesis are consistent with the standard axioms (such as the Zermelo-Fraenkel axioms) of set theory. After Nazi Germany annexed Austria on March 12, 1938, Gödel found himself in a rather awkward situation, partly because he had a long history of close associations with various Jewish members of the Vienna Circle (indeed, he had been attacked on the streets of Vienna by youths who thought that he was Jewish) and partly because he was suddenly in danger of being conscripted into the German army. Austrian-born mathematician, logician, and philosopher who obtained what may be the most important mathematical result of the 20th century: his famous incompleteness theorem, which states that within any axiomatic mathematical system there are propositions that cannot be proved or disproved on the basis of the axioms within that system; thus, such a system cannot be simultaneously complete and consistent. On Sept. 20, 1938, Gödel married Adele Nimbursky (née Porkert), and, when World War II broke out a year later, he fled Europe with his wife, taking the trans-Siberian railway across Asia, sailing across the Pacific Ocean, and then taking another train across the United States to Princeton, N.J., where, with the help of Einstein, he took up a position at the newly formed Institute for Advanced Studies (IAS).