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.
Cited by¶
12 citations across 12 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).
Supported in partVerified against the publisher's abstract
The fetched text of Gödel 1931 discusses the undecidable sentence obtained for Principia Mathematica and its arithmetic image, supporting the article's 'formal result' clause but not the general quantification or the second clause.
“Man beachte, daß “[ R (q); q ]” (oder was dasselbe bedeutet “[ S; q ]”) bloß eine metamathematische Beschreibung des unentscheidbaren Satzes ist.”
- The dramatic case is the formal result that any sufficiently rich consistent system contains undecidable statements; the pragmatic version is everywhere.
- 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).
Supported in partVerified against the publisher's abstract
Supplied abstract-level text of Gödel (1931) treats undecidable sentences and unprovable existence results in formal systems, backing the incompleteness frame but not the consistency/completeness/expressiveness trilemma as stated.
“die Existenz dieses Gegenbeispiels ist aber nach Satz IX nicht immer nachweisbar (in den angeführten formalen Systemen)”
- And in logic, Gödel's incompleteness shows consistency, completeness, and sufficient expressiveness are jointly unsatisfiable for a formal system.
- 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.
Supported in partVerified against the publisher's abstract
The paper's summary speaks of 'des unentscheidbaren Satzes' for the formal systems treated, but states neither the general recursively-axiomatised-theory result nor truth-but-unprovability.
“Man beachte, daß “[ R (q); q ]” (oder was dasselbe bedeutet “[ S; q ]”) bloß eine metamathematische Beschreibung des unentscheidbaren Satzes ist. Doch kann man, sobald man die Formel S ermittelt hat, natürlich auch die Zahl q bestimmen und damit den unentscheidbaren Satz selbst effektiv hinschreiben.”
- 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.
- 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.
Supported in partVerified against the publisher's abstract
Gödel 1931 is the incompleteness paper and its abstract mentions undecidable sentences, but it does not support the halting problem, Hilbert's tenth problem, or the same-self-reference-engine framing.
“Man beachte, daß “[ R (q); q ]” (oder was dasselbe bedeutet “[ S; q ]”) bloß eine metamathematische Beschreibung des unentscheidbaren Satzes ist.”
- 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
- 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.
Supported in partVerified against the publisher's abstract
Gödel 1931 is the work that introduced undecidable sentences in Principia Mathematica and related systems, but its abstract states no completeness theorem, so it grounds only the frame of this claim.
“die Existenz dieses Gegenbeispiels ist aber nach Satz IX nicht immer nachweisbar (in den angeführten formalen Systemen).”
- Gödel's result is that for expressive systems the two pull apart — a consistent 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.
Supported in partVerified against the publisher's abstract
Gödel 1931 is the canonical locus of the incompleteness theorems, but the packet holds only the publisher landing page and a fragment about an undecidable sentence.
“Man beachte, daß “[ R (q); q ]” (oder was dasselbe bedeutet “[ S; q ]”) bloß eine metamathematische Beschreibung des unentscheidbaren Satzes ist.”
- Structural tension: Gödel's incompleteness theorems
- 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.
Supported in partVerified against the publisher's abstract
German text with the DOI names the undecidable sentence Gödel constructs, but says nothing of self-unprovability or the diagonal lemma; frame only, no full text.
“Man beachte, daß “[ R (q); q ]” (oder was dasselbe bedeutet “[ S; q ]”) bloß eine metamathematische Beschreibung des unentscheidbaren Satzes ist. Doch kann man, sobald man die Formel S ermittelt hat, natürlich auch die Zahl q bestimmen und damit den unentscheidbaren Satz selbst effektiv hinschreiben.”
- In formal logic, Gödel's first incompleteness theorem constructs a sentence asserting its own unprovability, the diagonal lemma at its heart.
- 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.
Supported in partVerified against the publisher's abstract
Gödel 1931 takes formal systems as the objects of his undecidability results, but says nothing about "rules" or "code", and nothing about Church, Turing or Tarski.
“Die nachfolgenden Überlegungen gelten auch für die in den letzten Jahren von D. Hilbert und seinen Mitarbeitern aufgestellten formalen Systeme (soweit diese bisher vorliegen).”
- 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.
SupportedVerified against the publisher's abstract
Godel states that the procedure just described "liefert ein isomorphes Bild des Systems PM im Bereich der Arithmetik", so that metamathematical reasoning about that system can equally be carried out on the arithmetical image. Reproducing the formal system inside arithmetic is the arithmetization the sentence attributes to the 1931 paper. The passage states the arithmetization result without the term Godel numbering, or the word syntax, that the sentence's surrounding usage supplies.
“Das oben beschriebene Verfahren liefert ein isomorphes Bild des Systems PM im Bereich der Arithmetik und man kann alle metamathematischen Überlegungen ebenso gut an diesem isomorphen Bild vornehmen.”
- Gödel's arithmetization (1931
- 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¶
Mechanisms¶
- Metatheory Review Checklist
- Its strength is catching category errors — mistaking formal validity for real-world truth, legality, or safety — and surfacing intrinsic limits (incompleteness, undecidability)
This sourceEstablishes incompleteness and formally undecidable propositions for sufficiently strong formal systems.
- Its strength is catching category errors — mistaking formal validity for real-world truth, legality, or safety — and surfacing intrinsic limits (incompleteness, undecidability)
Verification¶
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Does it back the claim? Read against the text for 9 of 12 citations: 1 supported, 8 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
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