Peano arithmetic
E353625
Peano arithmetic is a formal first-order axiomatic system that captures the basic properties of the natural numbers and underpins much of modern mathematical logic and number theory.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Peano arithmetic canonical | 6 |
| Peano axioms | 4 |
| Peano arithmetic (under standard assumptions) | 1 |
| Peano axioms for natural numbers | 1 |
| first-order Peano arithmetic | 1 |
| second-order Peano arithmetic | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3380867 — 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: Peano arithmetic Context triple: [Tarski's undefinability theorem, appliesTo, Peano arithmetic]
-
A.
Gödel numbering
Gödel numbering is a method in mathematical logic that encodes symbols, formulas, and proofs as unique natural numbers, enabling arithmetic to represent and reason about syntactic statements.
-
B.
Lectures on the Logic of Arithmetic
Lectures on the Logic of Arithmetic is an educational work by Mary Everest Boole that explores the foundations and teaching of arithmetic through logical and psychological principles.
-
C.
Peano notation
Peano notation is a formal symbolic system for representing natural numbers and arithmetic operations using axioms and successor functions, developed by Giuseppe Peano.
-
D.
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.
-
E.
Hilbert’s second problem
Hilbert’s second problem is one of David Hilbert’s famous list of 23 problems, asking for a proof of the consistency of arithmetic from a finite set of axioms using finitary methods.
- 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: Peano arithmetic Target entity description: Peano arithmetic is a formal first-order axiomatic system that captures the basic properties of the natural numbers and underpins much of modern mathematical logic and number theory.
-
A.
Gödel numbering
Gödel numbering is a method in mathematical logic that encodes symbols, formulas, and proofs as unique natural numbers, enabling arithmetic to represent and reason about syntactic statements.
-
B.
Lectures on the Logic of Arithmetic
Lectures on the Logic of Arithmetic is an educational work by Mary Everest Boole that explores the foundations and teaching of arithmetic through logical and psychological principles.
-
C.
Peano notation
Peano notation is a formal symbolic system for representing natural numbers and arithmetic operations using axioms and successor functions, developed by Giuseppe Peano.
-
D.
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.
-
E.
Hilbert’s second problem
Hilbert’s second problem is one of David Hilbert’s famous list of 23 problems, asking for a proof of the consistency of arithmetic from a finite set of axioms using finitary methods.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
axiomatic system
ⓘ
first-order theory ⓘ formal system ⓘ mathematical logic theory ⓘ theory of arithmetic ⓘ |
| basedOn |
Peano arithmetic
self-linksurface differs
ⓘ
surface form:
Peano axioms
|
| captures | basic properties of the natural numbers ⓘ |
| doesNotQuantifyOver | sets of natural numbers ⓘ |
| field |
mathematical logic
ⓘ
number theory ⓘ |
| formalizes | induction on natural numbers ⓘ |
| hasAxiom |
0 is a natural number
ⓘ
0 is not the successor of any natural number ⓘ axioms defining addition recursively ⓘ axioms defining multiplication recursively ⓘ distinct natural numbers have distinct successors ⓘ every natural number has a unique successor ⓘ induction schema ⓘ |
| hasConsequence | basic theorems of elementary number theory ⓘ |
| hasConstantSymbol | 0 ⓘ |
| hasFeature | induction over all first-order formulas ⓘ |
| hasFunctionSymbol |
addition
ⓘ
multiplication ⓘ successor function ⓘ |
| hasModel |
nonstandard models of arithmetic
ⓘ
standard model of the natural numbers ⓘ |
| hasProperty |
consistent (if standard mathematics is consistent)
ⓘ
effectively axiomatizable ⓘ incomplete ⓘ recursively axiomatizable ⓘ undecidable ⓘ |
| hasRelationSymbol | equality ⓘ |
| hasVariant |
Peano arithmetic
self-linksurface differs
ⓘ
surface form:
first-order Peano arithmetic
Peano arithmetic self-linksurface differs ⓘ
surface form:
second-order Peano arithmetic
|
| impliedBy |
Gödel's incompleteness theorems
ⓘ
surface form:
Gödel incompleteness theorems
|
| introducedBy | Giuseppe Peano ⓘ |
| isStrongerThan | Robinson arithmetic ⓘ |
| isWeakerThan |
Zermelo–Fraenkel set theory
ⓘ
second-order arithmetic ⓘ |
| language | first-order language of arithmetic ⓘ |
| namedAfter | Giuseppe Peano ⓘ |
| quantifiesOver | individual natural numbers ⓘ |
| studiedIn |
model theory
ⓘ
proof theory ⓘ |
| underpins |
formal theories of computation
ⓘ
much of modern mathematical logic ⓘ |
| usedIn |
formalization of number theory
ⓘ
foundations of mathematics ⓘ |
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: Peano arithmetic Description of subject: Peano arithmetic is a formal first-order axiomatic system that captures the basic properties of the natural numbers and underpins much of modern mathematical logic and number theory.
Referenced by (14)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Peano axioms
this entity surface form:
Peano axioms
this entity surface form:
Peano arithmetic (under standard assumptions)
this entity surface form:
Peano axioms
this entity surface form:
first-order Peano arithmetic
this entity surface form:
second-order Peano arithmetic
this entity surface form:
Peano axioms for natural numbers
this entity surface form:
Peano axioms