Gödel's theorem(s)


Gödel's theorem(s)
Gödel's first incompleteness theorem states that for any consistent logical system S able to express arithmetic there must exist sentences that are true in the standard interpretation of S, but not provable. Moreover, if S is omega-consistent then there exist sentences such that neither they nor their negations are provable. The second theorem states that no such system can be powerful enough to prove its own consistency. These results, published in 1931 (‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’; trs. as ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems I’), determined the limits of purely formal methods in mathematics, and in particular marked the end of Hilbert's programme . Additional philosophical significance attaches to the way Gödel proved his first result. This was by defining a formula P that whilst unprovable can be seen to be true, given the way it is constructed. The implied moral is that truth in some way outruns provability, at least when that is considered formally.
Gödel proceeded by encoding logical formulae as numbers, so that to a statement about provability in the metatheory there corresponded by the mapping a simple statement of elementary mathematics. The formalism of arithmetic thus contains sentences that, considered via the enumeration, express propositions of their own metamathematics. The formula P is such a one: it expresses, to one interpreting it via the enumeration, its own unprovability. Gödel's procedure thus is an instance of a diagonal argument, and it bears a vivid analogy to examples of the semantic paradoxes . It is thus extremely important that all the methods he used are perfectly rigorous.
Gödel's procedure and his results opened the way to a fully rigorous treatment of the notion of a computable function, and to our modern understanding of the power and limits of computation, and the possibility or otherwise of programs that test for consistency and completeness.

Philosophy dictionary. . 2011.

Look at other dictionaries:

  • Gödel's theorem — n. either of two theorems published by the mathematician Kurt Gödel in 1931 that prove all mathematical systems are incomplete in that their truth or consistency can only be proved using a system of a higher order: also called Gödel s proof or… …   Universalium

  • Gödel's theorem — n. either of two theorems published by the mathematician Kurt Gödel in 1931 that prove all mathematical systems are incomplete in that their truth or consistency can only be proved using a system of a higher order: also called Gödel s proof or… …   English World dictionary

  • Gödel's theorem — may refer to: *Gödel s incompleteness theorems *Gödel s completeness theorem …   Wikipedia

  • gödel's theorem — noun also gödel s incompleteness theorem ˈgœ̅dəlz Usage: usually capitalized G Etymology: after Kurt Gödel died 1978 American mathematician : a theorem in advanced logic: in any logical system as complex or more complex than the arithmetic of the …   Useful english dictionary

  • Godel's theorem — noun Etymology: Kurt Gödel died 1978 American mathematician Date: 1933 a theorem in advanced logic: in any logical system as complex as or more complex than the arithmetic of the integers there can always be found either a statement which can be… …   New Collegiate Dictionary

  • Godel’s Theorem — all branches of mathematics are based on propositions that can’t be proved within that branch (named for mathematician Kurt Godel) …   Eponyms, nicknames, and geographical games

  • Gödel's theorem — /ˈgɜdəlz θɪərəm/ (say gerduhlz thearruhm) noun the proposition that in a formal axiomatic system, such as logic or mathematics, it is impossible to prove consistency without using methods beyond those of the system itself. {from Kurt Gödel,… …   Australian English dictionary

  • Gödel's theorem(s) — …   Philosophy dictionary

  • Gödel's incompleteness theorems — In mathematical logic, Gödel s incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. The theorems are of… …   Wikipedia

  • Gödel, Kurt — born April 28, 1906, Brünn, Austria Hungary died Jan. 14, 1978, Princeton, N.J., U.S. Austrian born U.S. mathematician and logician. He began his career on the faculty of the University of Vienna, where he produced his groundbreaking proof (see… …   Universalium


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