Skolem arithmetic
E865123
Skolem arithmetic is a fragment of first-order arithmetic focusing on the natural numbers with multiplication but without addition, studied for its distinctive decidability and model-theoretic properties.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Skolem arithmetic canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10462279 — 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: Skolem arithmetic Context triple: [Thoralf Skolem, notableWork, Skolem arithmetic]
-
A.
Peano arithmetic
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.
-
B.
Tarski–Mostowski–Robinson theorem
The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
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C.
Skolemization
Skolemization is a logical transformation technique that eliminates existential quantifiers by introducing Skolem functions or constants, commonly used in automated theorem proving and first-order logic.
-
D.
Herbrand's theorem
Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
-
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: Skolem arithmetic Target entity description: Skolem arithmetic is a fragment of first-order arithmetic focusing on the natural numbers with multiplication but without addition, studied for its distinctive decidability and model-theoretic properties.
-
A.
Peano arithmetic
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.
-
B.
Tarski–Mostowski–Robinson theorem
The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
-
C.
Skolemization
Skolemization is a logical transformation technique that eliminates existential quantifiers by introducing Skolem functions or constants, commonly used in automated theorem proving and first-order logic.
-
D.
Herbrand's theorem
Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
-
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 (44)
| Predicate | Object |
|---|---|
| instanceOf |
decidable theory
ⓘ
first-order theory ⓘ fragment of arithmetic ⓘ |
| allows | first-order formulas with multiplication and equality ⓘ |
| appearsIn | research on theories of arithmetic with restricted operations ⓘ |
| comparedTo | Presburger arithmetic as additive analogue NERFINISHED ⓘ |
| contrastWith |
Peano arithmetic
NERFINISHED
ⓘ
Presburger arithmetic NERFINISHED ⓘ |
| decidabilityReason | restricted language with multiplication only ⓘ |
| domainOfDiscourse | natural numbers ⓘ |
| excludesSymbol | addition ⓘ |
| field |
mathematical logic
ⓘ
model theory ⓘ proof theory ⓘ |
| hasModels |
nonstandard models of (ℕ, ×)
ⓘ
standard model (ℕ, ×) ⓘ |
| hasProperty |
complete
ⓘ
decidable ⓘ elimination of quantifiers in an appropriate language ⓘ model-complete ⓘ |
| historicalPeriod | 20th century development in mathematical logic ⓘ |
| influencedBy | early work of Thoralf Skolem on arithmetic theories ⓘ |
| isFragmentOf |
elementary arithmetic without addition
ⓘ
first-order arithmetic ⓘ |
| language |
first-order language with equality
ⓘ
first-order language with multiplication ⓘ |
| logicalStatus |
decidable by effective procedure
ⓘ
recursively axiomatizable ⓘ |
| namedAfter | Thoralf Skolem NERFINISHED ⓘ |
| quantifiesOver | individual variables for natural numbers ⓘ |
| relatedConcept |
Diophantine equations in multiplication only
ⓘ
monadic theories of arithmetic ⓘ theory of multiplication of natural numbers ⓘ |
| restriction |
no addition function symbol
ⓘ
no full induction schema for all formulas in richer languages ⓘ |
| studiedFor |
classification of definable sets in (ℕ, ×)
ⓘ
decidability properties ⓘ model-theoretic properties ⓘ |
| studiesStructure | (ℕ, ×) ⓘ |
| symbol | multiplication ⓘ |
| typicalResult |
classification of definable subsets of ℕ^k under multiplication
ⓘ
decidability of the first-order theory of (ℕ, ×) ⓘ |
| usedIn |
examples in model theory textbooks
ⓘ
investigations of the boundary between decidable and undecidable theories ⓘ |
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: Skolem arithmetic Description of subject: Skolem arithmetic is a fragment of first-order arithmetic focusing on the natural numbers with multiplication but without addition, studied for its distinctive decidability and model-theoretic properties.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.