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.
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 (symbol ?, from Peano's e) and implication (symbol ?, 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.[4] Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.[5]
