Hilbert-style deductive systems
E418216
Hilbert-style deductive systems are axiomatic proof systems in mathematical logic that use a small set of axiom schemas and a few inference rules (typically including modus ponens) to derive theorems in formal theories such as Zermelo–Fraenkel set theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Hilbert system for propositional logic | 1 |
| Hilbert-style deductive systems canonical | 1 |
| intuitionistic Hilbert system | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4165863 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hilbert-style deductive systems Context triple: [ZF, isFormalizedIn, Hilbert-style deductive systems]
-
A.
Recherches sur la théorie de la démonstration
Recherches sur la théorie de la démonstration is Jacques Herbrand’s foundational work in mathematical logic, introducing key results in proof theory and what is now known as Herbrand’s theorem.
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B.
New Foundations for Mathematical Logic
New Foundations for Mathematical Logic is W.V.O. Quine’s influential essay proposing an alternative set theory, known as "New Foundations," aimed at resolving paradoxes while preserving a broad, intuitive universe of sets.
-
C.
Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
-
D.
On a Problem of Formal Logic
"On a Problem of Formal Logic" is a seminal philosophical and mathematical paper by F. P. Ramsey that contributed to the foundations of logic and helped inspire what is now known as Ramsey theory.
-
E.
Tarski’s theorem on the completeness of elementary algebra and geometry
Tarski’s theorem on the completeness of elementary algebra and geometry is a foundational result in mathematical logic showing that the first-order theory of real closed fields (capturing elementary algebra and Euclidean geometry) is complete, decidable, and admits quantifier elimination.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hilbert-style deductive systems Target entity description: Hilbert-style deductive systems are axiomatic proof systems in mathematical logic that use a small set of axiom schemas and a few inference rules (typically including modus ponens) to derive theorems in formal theories such as Zermelo–Fraenkel set theory.
-
A.
Recherches sur la théorie de la démonstration
Recherches sur la théorie de la démonstration is Jacques Herbrand’s foundational work in mathematical logic, introducing key results in proof theory and what is now known as Herbrand’s theorem.
-
B.
New Foundations for Mathematical Logic
New Foundations for Mathematical Logic is W.V.O. Quine’s influential essay proposing an alternative set theory, known as "New Foundations," aimed at resolving paradoxes while preserving a broad, intuitive universe of sets.
-
C.
Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
-
D.
On a Problem of Formal Logic
"On a Problem of Formal Logic" is a seminal philosophical and mathematical paper by F. P. Ramsey that contributed to the foundations of logic and helped inspire what is now known as Ramsey theory.
-
E.
Tarski’s theorem on the completeness of elementary algebra and geometry
Tarski’s theorem on the completeness of elementary algebra and geometry is a foundational result in mathematical logic showing that the first-order theory of real closed fields (capturing elementary algebra and Euclidean geometry) is complete, decidable, and admits quantifier elimination.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
axiomatic proof system
ⓘ
deductive system ⓘ formal proof system ⓘ |
| allows | formal derivations from axioms ⓘ |
| appliesTo |
Peano arithmetic
ⓘ
Zermelo–Fraenkel set theory ⓘ first-order logic ⓘ formal theories ⓘ predicate logic ⓘ propositional logic ⓘ |
| associatedWith | David Hilbert ⓘ |
| characteristic |
each step justified by axiom or rule
ⓘ
finite or small set of axiom schemas ⓘ proofs as finite sequences of formulas ⓘ small number of inference rules ⓘ |
| component |
inference rules schema
ⓘ
logical axioms ⓘ nonlogical axioms of a theory ⓘ |
| contrastedWith |
natural deduction systems
ⓘ
sequent calculi ⓘ |
| feature |
axioms encode logical behavior
ⓘ
often use implication as primitive connective ⓘ other connectives defined via axioms ⓘ rules are few and simple ⓘ |
| field | mathematical logic ⓘ |
| goal | derive theorems ⓘ |
| hasVariant |
Hilbert-style deductive systems
self-linksurface differs
ⓘ
surface form:
Hilbert system for propositional logic
Hilbert-style deductive systems self-linksurface differs ⓘ
surface form:
intuitionistic Hilbert system
modal Hilbert system ⓘ |
| historicalOrigin | Hilbert’s program ⓘ |
| inferenceRule |
generalization rule
ⓘ
modus ponens ⓘ |
| property |
complete for many standard logics
ⓘ
sound with respect to standard semantics when well-formed ⓘ |
| typicalAxiomForm |
implication axioms
ⓘ
quantifier axioms ⓘ |
| typicallyUses | modus ponens ⓘ |
| usedFor |
completeness proofs
ⓘ
consistency proofs ⓘ formalization of mathematics ⓘ metalogical investigations ⓘ proof theory ⓘ soundness proofs ⓘ |
| usedIn |
classical first-order logic
ⓘ
formalization of Zermelo–Fraenkel set theory ⓘ formalization of first-order arithmetic ⓘ |
| uses |
axiom schemas
ⓘ
inference rules ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hilbert-style deductive systems Description of subject: Hilbert-style deductive systems are axiomatic proof systems in mathematical logic that use a small set of axiom schemas and a few inference rules (typically including modus ponens) to derive theorems in formal theories such as Zermelo–Fraenkel set theory.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
Hilbert-style deductive systems
→
hasVariant
→
Hilbert-style deductive systems
self-linksurface differs
ⓘ
this entity surface form:
intuitionistic Hilbert system
Hilbert-style deductive systems
→
hasVariant
→
Hilbert-style deductive systems
self-linksurface differs
ⓘ
this entity surface form:
Hilbert system for propositional logic