Construct · 1 fields · 3 citations

Gödel's incompleteness theorems: a standalone construct

One name. One construct. A landmark on the map.

● anchor · a standalone construct Nearest construct by meaningconsistencycompare the meanings
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Every construct on the map is compared with every other by the meaning of its definitions, and the closest one is surfaced here. The nearest construct to Gödel's incompleteness theorems is consistency.

Hold this as a prompt, not a verdict. Two constructs sitting close in meaning is a reason to read them side by side and decide for yourself whether they are one idea under two names. It is not proof that they are. Compare the meanings in the Collider.

Gödel's incompletenesstheorems1 CONSTRUCTIncompletenessGÖDEL · INCOMPLETENESS

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Mathematical logic

Limits on what a consistent formal system that expresses arithmetic can prove

Kurt Gödel's 1931 paper Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I proved two results about formal systems, the fixed rule-bound frameworks in which mathematics is written as strings of symbols and derived by mechanical steps. The first incompleteness theorem shows that any such system that is consistent (never proves both a statement and its negation) and strong enough to express basic arithmetic must contain true statements about the whole numbers that the system itself can neither prove nor disprove. The second incompleteness theorem shows that such a system cannot prove its own consistency using only its own methods. Gödel's construction encodes statements about proofs as statements about numbers, then builds a sentence that in effect asserts its own unprovability, so the sentence turns out true exactly when the system cannot prove it. J. Barkley Rosser's 1936 extension weakened the technical assumption Gödel had needed, and the theorems stand as a fixed limit on what any single consistent formal axiom system can settle about arithmetic, separate from whether a given true statement can be reached from some stronger system.[1][2][3]

Unit
A pair of theorems about the provability limits of consistent formal systems that can express arithmetic
Recovery means
Not applicable
Measured by
Established by formal proof: arithmetization of syntax (Gödel numbering) and construction of a self-referential sentence the system can neither prove nor disprove

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Nearby constructs

consistency4 fields · 5 citationsjingle · one word, four constructshalting problem1 fields · 2 citationsanchor · a standalone constructproof2 fields · 5 citationsjingle · one word, two constructs

References

  1. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38(1), 173–198. https://doi.org/10.1007/BF01700692
  2. Rosser, B. (1936). Extensions of some theorems of Gödel and Church. The Journal of Symbolic Logic, 1(3), 87–91. https://doi.org/10.2307/2269028
  3. Smith, P. (2013). An introduction to Gödel's theorems (2nd ed.). Cambridge University Press.

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