Hilbert-style deductive systems

E418216

axiomatic proof system deductive system formal proof system

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.

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All labels observed (6)

Label	Occurrences
Hilbert system for propositional logic	1
Hilbert systems	1
Hilbert-style deductive systems canonical	1
Hilbert-style proof systems	1
Hilbert-style systems	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.
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 system for propositional logic ⓘ linked to: Hilbert-style deductive systems intuitionistic Hilbert system ⓘ linked to: Hilbert-style deductive systems 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