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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.