Peano notation
E13831
Peano notation is a formal symbolic system for representing natural numbers and arithmetic operations using axioms and successor functions, developed by Giuseppe Peano.
Aliases (2)
- Peano arithmetic ×1
- Peano axioms ×1
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
formal system
→
mathematical notation → representation of natural numbers → |
| assumes |
existence of a distinguished element 0
→
existence of a unary successor operation → |
| basedOn |
Peano axioms
→
|
| category |
symbolic representation system
→
|
| clarifies |
structure of natural number system
→
|
| defines |
successor of a natural number
→
zero as the first natural number → |
| developedBy |
Giuseppe Peano
→
|
| enables |
formal proofs about arithmetic properties
→
|
| ensures |
0 is not the successor of any number
→
different numbers have different successors → uniqueness of successor → |
| exampleRepresentation |
1 is represented as S(0)
→
2 is represented as S(S(0)) → 3 is represented as S(S(S(0))) → |
| field |
mathematical logic
→
number theory → |
| formalizes |
addition
→
induction on natural numbers → multiplication → |
| goal |
eliminate ambiguity in arithmetic reasoning
→
provide rigorous foundation for natural numbers → |
| historicalPeriod |
late 19th century
→
|
| influenced |
formal languages for arithmetic
→
modern axiomatic set theory → |
| introducedIn |
axiomatization of arithmetic by Peano
→
|
| language |
symbolic mathematical language
→
|
| notationStyle |
unary numeral system
→
|
| relatedTo |
Peano arithmetic
→
first-order arithmetic → |
| represents |
arithmetic operations
→
natural numbers → |
| supports |
definition of recursive functions on natural numbers
→
|
| symbolForSuccessor |
S
→
|
| symbolForZero |
0
→
|
| usedIn |
formal verification of arithmetic
→
foundations of arithmetic → proof theory → |
| usesConcept |
axiomatic method
→
successor function → |
Referenced by (3)
| Subject (surface form when different) | Predicate |
|---|---|
|
Peano notation
("Peano axioms")
→
|
basedOn |
|
Principia Mathematica
→
|
influencedBy |
|
Peano notation
("Peano arithmetic")
→
|
relatedTo |