Formal Theories - Formal Systems, Formal Theories of Arithmetic, Axiom, Peano Axioms, Presburger Arithmetic, Second-Order Arithmetic (Paperback)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 32. Chapters: Formal systems, Formal theories of arithmetic, Axiom, Peano axioms, Presburger arithmetic, Second-order arithmetic, Physical symbol system, Undecidable problem, Theory, Axiomatic system, Formal ethics, Robinson arithmetic, Rule of inference, True arithmetic, Proof, Set-theoretic definition of natural numbers, Primitive recursive arithmetic, Typographical Number Theory, Non-standard model of arithmetic, Formal proof, Axiom schema, Deductive system, Theory of pure equality. Excerpt: 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. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed a collection of axioms about the numbers, 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: ). The Peano axioms contain three types of statements. The first axiom asserts the existence of a least one member of the set "number." The next four are general statements about equality; in modern treatments these are often considered axioms of the "underlying logic." The next three axioms are first-order statements about natural numbers expressing the fundamental properti...