Mathematical axioms | Logic in computer science | Mathematical logic | Formal theories of arithmetic
In mathematical logic, the Peano axioms, also known as the DedekindâPeano axioms or the Peano postulates, are 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. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified 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). The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. 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". 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, which makes this formulation close to second-order arithmetic. 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. (Wikipedia).
RA1.3. Peano Axioms and Induction
Real Analysis: We consider the Peano Axioms, which are used to define the natural numbers. Special attention is given to Mathematical Induction and the Well-Ordering Principle for N. (Included is an example of how to show a triple equivalence.)
From playlist Real Analysis
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From playlist Set Theory by Mathoma
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From playlist Set Theory by Mathoma
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From playlist Course 6: Introduction to Analysis (Fall 2017)
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From playlist Summer of Math Exposition Youtube Videos
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Viewers like you help make PBS (Thank you đ) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi If you needed to tell someone what numbers are and how they work, without using the notion of number in your answer, could you do it? Tweet at us! @pbsinfinite Faceboo
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