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# Peano axioms, also known as Peano’s postulates, in number theory, five axioms introduced in 1889 by Italian mathematician Giuseppe Peano.<ref name="ref_d5b36444">[https://www.britannica.com/science/Peano-axioms Peano axioms | mathematics]</ref> | # Peano axioms, also known as Peano’s postulates, in number theory, five axioms introduced in 1889 by Italian mathematician Giuseppe Peano.<ref name="ref_d5b36444">[https://www.britannica.com/science/Peano-axioms Peano axioms | mathematics]</ref> |
2021년 2월 25일 (목) 23:45 기준 최신판
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말뭉치
- Peano axioms, also known as Peano’s postulates, in number theory, five axioms introduced in 1889 by Italian mathematician Giuseppe Peano.[1]
- The chapter demonstrates that this theorem, while true, is not provable in Peano arithmetic.[2]
- Paris and Harrington (1977) gave the first "natural" example of a statement which is true for the integers but unprovable in Peano arithmetic (Spencer 1983).[3]
- Peano arithmetic refers to a theory which formalizes arithmetic operations on the natural numbers ℕ \mathbb{N} and their properties.[4]
- Peano's axioms are the basis for the version of number theory known as Peano arithmetic.[5]
- We show that a cyclic formulation of first-order arithmetic is equivalent in power to Peano Arithmetic.[6]
- Our work will result in the so-called Peano axioms.[7]
- The modern version of Peano axioms can be put as follows.[7]
- Traditionally, the first order arithmetic is called Peano arithmetic, and is denoted simply by PA (instead of the above PA 2 ).[7]
- The nine Peano axioms contain three types of statements.[8]
- The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".[8]
- Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.[8]
- In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a recursively enumerable set of axioms.[8]
- The system of Peano arithmetic in first-order language, mentioned at the end of the article, is no longer categorical (cf.[9]
- And in 1891 Giuseppe Peano gave essentially the Peano axioms listed here (they were also given slightly less formally by Richard Dedekind in 1888)—which have been used unchanged ever since.[10]
- The proof of Gödel's Theorem in 1931 (see page 1158) demonstrated the universality of the Peano axioms.[10]
- The Peano axioms for arithmetic seem sufficient to support most of the whole field of number theory.[10]
- The first contribution of this paper is answering this question for infinitary Peano arithmetic.[11]
- Thus, non-commutative infinitary Peano arithmetic is shown to be a subclassical logic.[11]
- The second contribution of this paper is introducing infinitary Peano arithmetic having antecedent-grouping and no right exchange rules.[11]
- Some historians insist on using the term “Dedekind-Peano axioms”.[12]
소스
- ↑ Peano axioms | mathematics
- ↑ A Mathematical Incompleteness in Peano Arithmetic *
- ↑ Peano Arithmetic -- from Wolfram MathWorld
- ↑ Peano arithmetic in nLab
- ↑ Peano's Axioms -- from Wolfram MathWorld
- ↑ Cyclic Arithmetic Is Equivalent to Peano Arithmetic
- ↑ 7.0 7.1 7.2 Peano Axioms. First Order Arithmetic. By K.Podnieks
- ↑ 8.0 8.1 8.2 8.3 Peano axioms
- ↑ Encyclopedia of Mathematics
- ↑ 10.0 10.1 10.2 Note (a) for Implications for Mathematics and Its Foundations: A New Kind of Science
- ↑ 11.0 11.1 11.2 Non-Commutative Infinitary Peano Arithmetic
- ↑ (PDF) Peano and the Foundations of Arithmetic
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- [{'LOWER': 'peano'}, {'LEMMA': 'arithmetic'}]