Peano axioms (nonfiction): Difference between revisions
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The need to formalize [[Arithmetic (nonfiction)|arithmetic]] was not well appreciated until the work of [[Hermann Grassmann (nonfiction)|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 (nonfiction)|Charles Sanders Peirce]] provided an axiomatization of natural-number arithmetic. In 1888, [[Richard Dedekind (nonfiction)|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 ''[[Arithmetices principia, nova methodo exposita (nonfiction)|Arithmetices principia, nova methodo exposita]]'' ("The principles of arithmetic presented by a new method"). | The need to formalize [[Arithmetic (nonfiction)|arithmetic]] was not well appreciated until the work of [[Hermann Grassmann (nonfiction)|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 (nonfiction)|Charles Sanders Peirce]] provided an axiomatization of natural-number arithmetic. In 1888, [[Richard Dedekind (nonfiction)|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 ''[[Arithmetices principia, nova methodo exposita (nonfiction)|Arithmetices principia, nova methodo exposita]]'' ("The principles of arithmetic presented by a new method"). | ||
The 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 (nonfiction)|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 logic (nonfiction)|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 (nonfiction)|second-order induction]] axiom with a first-order axiom schema. | The 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 (nonfiction)|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 logic (nonfiction)|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 (nonfiction)|second-order induction]] axiom with a first-order [[Axiom schema (nonfiction)|axiom schema]]. | ||
== In the News == | == In the News == | ||
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== Fiction cross-reference == | == Fiction cross-reference == | ||
* [[Algorithmic Protocol Treaty Organization]] - APTO | |||
* [[Axiom Antics]] - rogue stand-up mathematical comedy and alleged front for organized [[transcorporate crime]]. | |||
* [[Crimes against mathematical constants]] | * [[Crimes against mathematical constants]] | ||
* [[Gnomon algorithm]] | * [[Gnomon algorithm]] | ||
Revision as of 06:50, 21 April 2020
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano 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 Arithmetices principia, nova methodo exposita ("The principles of arithmetic presented by a new method").
The 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. 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.
In the News
Fiction cross-reference
- Algorithmic Protocol Treaty Organization - APTO
- Axiom Antics - rogue stand-up mathematical comedy and alleged front for organized transcorporate crime.
- Crimes against mathematical constants
- Gnomon algorithm
- Gnomon Chronicles
- Mathematics
Nonfiction cross-reference
- Completeness - converse of Soundness (nonfiction)
- Mathematical logic (nonfiction)
- Mathematics (nonfiction)
- The principles of arithmetic presented by a new method (nonfiction)
External links
- Peano axioms @ Wikipedia