|
gptkbp:instanceOf
|
axiomatic system
|
|
gptkbp:alsoKnownAs
|
first-order arithmetic
|
|
gptkbp:basedOn
|
gptkb:Peano_axioms
|
|
gptkbp:describes
|
natural numbers
|
|
gptkbp:field
|
gptkb:logic
gptkb:mathematics
number theory
|
|
gptkbp:formalizes
|
addition
multiplication
|
|
gptkbp:hasApplication
|
gptkb:logic
computer science
foundations of mathematics
automated theorem proving
formal verification
|
|
gptkbp:hasAxiom
|
distinct natural numbers have distinct successors
every natural number has a unique successor
induction axiom
zero is a natural number
zero is not the successor of any natural number
|
|
gptkbp:hasInductionSchema
|
true
|
|
gptkbp:hasModel
|
gptkb:standard_model_of_natural_numbers
|
|
gptkbp:hasNonstandardModels
|
true
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
Peano arithmetic
|
|
gptkbp:introduced
|
gptkb:Giuseppe_Peano
|
|
gptkbp:introducedIn
|
1889
|
|
gptkbp:isConsistent
|
undecidable (by Gödel's incompleteness theorems)
|
|
gptkbp:isCountableTheory
|
true
|
|
gptkbp:isDecidable
|
false
|
|
gptkbp:isFinitelyAxiomatizable
|
false
|
|
gptkbp:isFoundationFor
|
arithmetic
elementary number theory
|
|
gptkbp:isIncomplete
|
true
|
|
gptkbp:isRecursivelyEnumerable
|
true
|
|
gptkbp:language
|
0, S, +, ×, = (zero, successor, addition, multiplication, equality)
|
|
gptkbp:relatedTo
|
gptkb:Gödel's_incompleteness_theorems
gptkb:Presburger_arithmetic
gptkb:Robinson_arithmetic
gptkb:second-order_arithmetic
|
|
gptkbp:studiedIn
|
model theory
proof theory
|
|
gptkbp:usesLogic
|
gptkb:first-order_logic
|
|
gptkbp:bfsParent
|
gptkb:Gödel's_incompleteness_theorems
gptkb:Gödel's_incompleteness_theorems_(1931)
gptkb:Tarski's_theorem
gptkb:Axiom_of_Infinity
|
|
gptkbp:bfsLayer
|
5
|