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In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.〔Grassmann 1861〕 In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic.〔Peirce 1881; also see Shields 1997〕 In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ((ラテン語:Arithmetices principia, nova methodo exposita)). The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set "number". 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".〔van Heijenoort 1967:94〕 The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema. == Formulation == When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (∈, which comes from Peano's ε) and implication (⊃, which comes from Peano's reversed 'C'.) Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the Begriffsschrift by Gottlob Frege, published in 1879.〔Van Heijenoort 1967, p. 2〕 Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.〔Van Heijenoort 1967, p. 83〕 The Peano axioms define the arithmetical properties of ''natural numbers'', usually represented as a set ''N'' or The signature (a formal language's non-logical symbols) for the axioms includes a constant symbol 0 and a unary function symbol ''S''. The constant 0 is assumed to be a natural number: The next four axioms describe the equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments.〔Van Heijenoort 1967, p. 83〕 The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" function ''S''. Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number.〔Peano 1889, p. 1〕 This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1 and 6 define a unary representation of the natural numbers: the number 1 can be defined as ''S''(0), 2 as ''S''(''S''(0)) (which is also ''S''(1)), and, in general, any natural number ''n'' as the result of ''n''-fold application of ''S'' to 0, denoted as ''S''''n''(0). The next two axioms define the properties of this representation. Axioms 1, 6, 7 and 8 imply that the set of natural numbers contains the distinct elements 0, ''S''(0), ''S''(''S''(0)), and furthermore that ⊆ ''N''. This shows that the set of natural numbers is infinite. However, to show that ''N'' = , it must be shown that ''N'' ⊆ ; i.e., it must be shown that every natural number is included in . To do this however requires an additional axiom, which is sometimes called the ''axiom of induction''. This axiom provides a method for reasoning about the set of all natural numbers. The induction axiom is sometimes stated in the following form: In Peano's original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section Models below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Peano axioms」の詳細全文を読む スポンサード リンク
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