Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.

Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38, 173-198.

Type

Journal article

Intellectual base

Primary research

Year

1931

DOI

10.1007/bf01700692

Link

https://doi.org/10.1007/BF01700692

Cited from

computer_science, linguistics_semiotics, mathematics, philosophy, systems_cybernetics

Cited by

11 citations across 11 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Axiom
  • The dramatic case is the formal result that any sufficiently rich consistent system contains undecidable statements; the pragmatic version is everywhere.

    This sourceProves that any sufficiently rich consistent formal system contains statements it can neither prove nor refute (the incompleteness theorems).

• Axiomatic Incompatibility
  • And in logic, Gödel's incompleteness shows consistency, completeness, and sufficient expressiveness are jointly unsatisfiable for a formal system.

    This sourceProves consistency, completeness, and sufficient expressiveness are jointly unsatisfiable for a formal system rich enough to encode arithmetic (the incompleteness theorems).

• Completeness
  • Gödel's 1931 incompleteness theorems exhibit, for any consistent recursively-axiomatised theory extending Peano arithmetic, a true-but-unprovable formula — establishing essential structural limits to deductive completeness.

    This sourceEstablishes the first and second incompleteness theorems for any consistent recursively-axiomatised theory extending a sufficient fragment of arithmetic, exhibiting a true-but-unprovable formula.

• Computability
  • Mathematics and logic. The halting problem; Hilbert's tenth (no algorithm decides Diophantine solvability); Gödel's incompleteness as a sibling result driven by the same self-reference engine

    This sourceThe incompleteness theorems, a sibling result driven by the same self-reference/diagonalization engine as undecidability.

• Consistency
  • Gödel's result is that for expressive systems the two pull apart — a consistent system cannot be complete.

    This sourceProves that a sufficiently expressive consistent formal system cannot be complete.

• Deductive Reasoning
  • Structural tension: Gödel's incompleteness theorems

    This sourceEstablishes the first and second incompleteness theorems: any consistent, sufficiently expressive formal system contains true statements unprovable from its axioms. WebSearch confirmed venue, pages, and theorem content.

• Diagonal Impossibility
  • In formal logic, Gödel's first incompleteness theorem constructs a sentence asserting its own unprovability, the diagonal lemma at its heart.

    This sourceFirst and second incompleteness theorems via the diagonal lemma; a system cannot prove its own consistency.

• Formal System
  • A formal system is sharper than "rules" or "code," which may be informal, contradictory, or judgment-laden; it is the artifact that the process of formalization aims at, and it is the substrate on which the meta-theorems of Gödel, Church, Turing, and Tarski operate.
• Meta-Symbolic Reflection
  • Gödel's arithmetization (1931

    This sourceEstablishes the first and second incompleteness theorems for any consistent recursively-axiomatised theory extending a sufficient fragment of arithmetic.

• Reflexivity (Self-Reference)

  • This sourceEstablishes the first and second incompleteness theorems for any consistent recursively-axiomatised theory extending a sufficient fragment of arithmetic.

Domain-specific

• Gödel numbering
  • Gödel numbering arithmetizes metasyntax: sequences receive codes, syntactic predicates become computable or representable number-theoretic relations, and a formal theory can express claims about its own formulas and proofs.

Verification

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