devutilsFrançais · article non traduitGitHub A result about formal theories
Gödel's article appeared in 1931. The Stanford survey places it in a history of foundational work and explains the assumptions behind incompleteness. These results concern provability in sufficiently strong effective formal theories, not a blanket claim that mathematics is contradictory or that no proof can ever succeed.
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Gödel's original argument used a stronger consistency assumption than the ordinary consistency appearing in later strengthened formulations. The second theorem also depends on the theory and its formalization of consistency. Those qualifications matter when explaining what the results establish.
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Read Birth of computer science for the distinction between consistency, completeness and effective decision procedures. The event uses year precision for the published work. It does not assign the publication date to every idea developed during the preceding research or equate incompleteness with the separate decision-problem results of Church and Turing.
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The 1931 publication limits what suitable formal arithmetic theories can prove, rather than declaring all reasoning impossible.
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- Stanford Encyclopedia of Philosophy, Gödel’s Incompleteness Theorems — Introduction; section 2.1; section 5, publication in January 1931
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- Stanford Encyclopedia of Philosophy, Gödel’s Incompleteness Theorems — Introduction; section 2.1; section 5, publication in January 1931
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