Austrian-American logician whose 1931 incompleteness theorems are among the most significant results in the foundations of mathematics: any consistent formal axiomatic system powerful enough to describe basic arithmetic contains true statements it cannot prove, and such a system cannot prove its own consistency from within itself. The result answered, negatively, the most ambitious form of David Hilbert's program to place all mathematics on one finite, provably consistent foundation. Godel emigrated to the United States in 1940, working at the Institute for Advanced Study in Princeton alongside Einstein, and became increasingly consumed by paranoid fear of being poisoned in his final years, eventually starving himself to death after his wife, who prepared all his food, was hospitalized.
Facts
BirthplaceBrno, Austria-Hungary (present-day Czech Republic) 1 Death PlacePrinceton, New Jersey, United States, of self-starvation 1 Nationality / CultureAustrian, later American citizen (naturalized 1948) 1 Defining ContributionThe incompleteness theorems (1931), showing fundamental limits on what any sufficiently powerful consistent formal system can prove about itself. 1 Notable WorkUeber formal unentscheidbare Saetze der Principia Mathematica und verwandter Systeme I (1931) 1 Connections
Associated With
Godel proved in 1940 that the negation of CH cannot be proved from standard ZFC set theory, the first half of the independence result completed by Cohen in 1963 (Cohen has no live entity in this atlas to link to).
Source Continuum Hypothesis (Wikipedia)
Source MacTutor History of Mathematics Archive
The incompleteness theorems set a permanent limit on what proof, as a method, can achieve for its own foundations.
Source The Stanford Encyclopedia of Philosophy
In Branch
Source The Stanford Encyclopedia of Philosophy
Namesake Of
Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)
Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)
Proofs Credited
Why this is disputed. Godel used one specific instance of this lemma in his 1931 incompleteness paper without naming it as a separate result; Rudolf Carnap, who is not yet a live mathematician entity in this atlas, published the lemma at a general level in 1934, so crediting sole proof to Godel is disputed rather than settled.
Source Diagonal Lemma (Wikipedia)
Source The Stanford Encyclopedia of Philosophy
Additional Source Wikipedia: Godel's Incompleteness TheoremsLead section
Sources
1. The Stanford Encyclopedia of Philosophy
Center for the Study of Language and Information, Stanford University
Wikidata: Kurt Godel
MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsAssociated With: Logic and Foundations, https://mathshistory.st-andrews.ac.uk/Biographies/Godel/Quote, Associated With: Logic and Foundations, https://mathshistory.st-andrews.ac.uk/Biographies/Godel/
He proved fundamental results about axiomatic systems, showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system.
Continuum Hypothesis (Wikipedia)
Wikimedia FoundationAssociated With: Continuum Hypothesis, Independence from ZFC section, second paragraphQuote, Associated With: Continuum Hypothesis, Independence from ZFC section, second paragraph
Kurt Godel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory.
Wikipedia: Godel's Incompleteness Theorems
Wikimedia FoundationProofs Credited: Godel's Incompleteness Theorems, Lead sectionQuote, Proofs Credited: Godel's Incompleteness Theorems, Lead section
These results, published by Kurt Godel in 1931, are important both in mathematical logic and in philosophy of mathematics.
Diagonal Lemma (Wikipedia)
Wikimedia FoundationProofs Credited: Diagonal Lemma, History sectionQuote, Proofs Credited: Diagonal Lemma, History section
In 1934, Rudolf Carnap was the first to publish the diagonal lemma in some level of generality
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