# Peano's Axiomatic System

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## References

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### 2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Natural_number#Peano_axioms
• The Peano axioms give a formal theory of the natural numbers starting with 0. The axioms are:
• There is a natural number 0.
• Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a+1.
• There is no natural number whose successor is 0.
• Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
• If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.)

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Peano_arithmetic
• 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 consistency and completeness of number theory.
• The need for formalism in 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. [1] In 1888, Richard Dedekind proposed a collection of axioms about the numbers,[2] 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 (Latin: Arithmetices principia, nova methodo exposita). [3]
• The Peano axioms contain three types of statements. The first four statements are general statements about equality; in modern treatments these are often considered axioms of pure logic. The next four 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 replacing this second-order induction axiom with a first-order axiom schema.
• ...
• The Peano axioms define the properties of natural numbers, usually represented as a set N or \mathbb{N}. The signature for the axioms includes a constant symbol 0 and a unary function symbol S.
• The first four axioms describe the equality relation. [6]
• 1. For every natural number x, x = x. That is, equality is reflexive.
• 2. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
• 3. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
• 4. For all a and b, if a is a natural number and a = b, then b is also a natural number. That is, the natural numbers are closed under equality.
• The remaining axioms define the properties of the natural numbers. The constant 0 is assumed to be a natural number, and the naturals are assumed to be closed under a "successor" function S.
• 5. 0 is a natural number.
• 6. For every natural number [math]n[/math], S(n) is a natural number.
• Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 5 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 5 and 6 define a unary representation of the natural numbers: the number 1 is S(0), 2 is S(S(0)) (= S(1)), and, in general, any natural number n is Sn(0). The next two axioms define the properties of this representation.
• 7. For every natural number [math]n[/math], S(n) ≠ 0. That is, there is no natural number whose successor is 0.
• 8. For all natural numbers m and [math]n[/math], if S(m) = S(n), then m = n. That is, S is an injection.
• These two axioms together imply that the set of natural numbers is infinite, because it contains at least the infinite subset { 0, S(0), S(S(0)), … }, each element of which differs from the rest. The final axiom, sometimes called the axiom of induction, is a method of reasoning about all natural numbers; it is the only second order axiom.
• 9. If K is a set such that:
• 0 is in K, and
• for every natural number [math]n[/math], if n is in K, then S(n) is in K, then K contains every natural number.